Download Chapter 2. Electronic, Vibrational and Spin

Chapter 2. Electronic, Vibrational and Spin Configurations of Electronically
Excited States.
2.1
Visualization of the Electronically Excited Structures in the Paradigms of
Molecular Organic Photochemistry
From the working paradigms of Sch. 1.1, 1.2 and 1.3, we have learned that
electronically excited states, (*R), are critical transient structures in photochemical
reactions. Any chemical process that is produced directly from R (*R → Ι, *I, *P, F) is
termed a primary photochemical process. Any physical process that proceeds directly
from R (*R → R) is termed a primary photophysical process. The most common
primary photochemical process for organic molecules is the *R → I as shown in Sch.
2.1. The latter is a working paradigm with the addition of the primary photophysical
processes *R → R (+ heat or + hν). These photophysical pathways will generally
compete with any primary photochemical pathways (Sch. 2.1) that take *R to P, i.e., the
primary photochemical process which proceeds through a ground state reactive
intermediate I to form P (*R → I → P, shown). For simplicity, the primary
photochemical process of Sch. 1.1 that takes *R to P through a funnel without formation
of a reactive intermediate (*R →F → P) and the primary photochemical process that
takes *R to P by way of electronically excited intermediates (*R →*I, *P → P) are not
shown.
Photochemical
R + hν
*R
P
I
Photophysical
Thermal
Scheme 2.1. The working paradigm for photochemical pathways of *R → P and
photophysical pathways from *R → R. See Scheme 1.1 for a more complete
paradigm.
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The reactive intermediate(s) I undergoes conventional thermal processes (I → P)
leading to the observed product, P. The structure I represents the common reactive
intermediates (radical pairs, diradical, carbonium ion/carbanion pairs, zwitterionic
species, carbene, etc.). The typical electron structure of I will be discussed in Chapter 6.
The thermal pathways from I → P are termed secondary processes and will be discussed
in detail in Chapter 8. “Direct” primary photochemical pathways from *R → F → P and
pathways involving formation of electronically excited state of product (e.g., *R → *I and
*R → *P) are known, but turn out to be rare experimentally, and will not be covered
explicitly until Chapter 6. The symbols (R, *R, I, P) given in Sch. 2.1 should not be
restricted to imply the involvement of single molecular species. For example, it is
intended that *R symbolizes the system of reactants which would include any other
molecules which are involved in the primary photochemical process. All of the species
in Sch. 2.1, including *R, can be represented by traditional Lewis structures and can be
discussed in terms of conventional one electron molecular orbitals (MO). It is assumed
that the student is familiar with Lewis structural and MO representations of R, P and I, all
of which possess the ordinary organic functional groups contained in organic molecules
and which discussed in all elementary courses in organic chemistry. It is a goal of the text
to make the student comfortable with applying knowledge of the structures of R, P and I
towards an understanding of the detailed electronic, vibrational and spin structure of *R.
A detailed understanding of primary photophysical and photochemical processes
is essential for a proper interpretation and prediction of the plausibility (is a process
implied by the paradigm plausible or not?) and probability (how does a given process
compete with other processes?) of the individual steps given by Sch. 2.1. An
understanding of all these processes is available through the paradigms of the molecular
structure and dynamics of organic molecules. Quantum mechanics provides the most
powerful and effective paradigm of physics and chemistry for the description of
molecular structure and dynamics of organic molecules. However, a proper
understanding of molecular structure and dynamics through quantum mechanics requires
a mathematical sophistication that is well beyond the intended scope and intended
readership of this text. This Chapter and Chapter 3 are concerned with the exploitation of
the power of quantum mechanical methods by translating the mathematics, which are
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required for a rigorous quantum mechanical treatment of organic photochemistry, into
pictures and structures that can be more readily comprehended by students who may not
have a background or interest in formal mathematics. We will take a pictorial approach
which has proven to be very effective: we shall translate the mathematics of quantum
mechanics into pictures that can be translated into molecular structures (this Chapter)
and molecular dynamics (Chapter 3). This pictorial approach recognizes that the
mathematical ideas of quantum mechanics can be interpreted in a qualitative manner
through pictorial representation and that these representations are understood to always to
a certain degree acceptable approximations.
The language of organic chemistry employs pictorial objects termed molecular
structures to make connections to molecular function, properties and reactivity. The
language of quantum mechanics uses mathematical objects termed wavefunctions (a
term we shall deal with in detail in this Chapter) and the idea of quantized energies to
make connections to molecular function, properties and reactivity. Since the molecule is
the same object being described by both molecular structures and wavefunctions it is
natural to expect that there are methods to transform molecule structures into
wavefunctions and to transform wavefunctions into molecular structure. Thus in Sch.
2.1, R, *R, I and P may be considered as ordinary molecular structures or as
wavefunctions. Visualizations of wavefunctions of R, *R, I and P, although understood to
be only approximate models, allow the photochemist’s imagination to create a "motion
picture" of the choreography of structural, energetic and dynamic events occurring at the
molecular level for the overall photochemical transformation of *R → P.
As in all analogies and models, such visualizations of wavefunctions are
incomplete and an imperfect representation of the more "correct" mathematics of
quantum mechanics, but such pictures are valid and useful approximations for
introducing a student to the concepts of a new field. Indeed, if the pictures provided
capture the correct nature of the fundamental structures, the fundamental forces and
realistic dynamics of the processes of Sch. 2.1, the student will achieve a valuable
introductory and operational knowledge of a field that also will be very helpful as a basis
for probing deeper into the subject.
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The visualization and pictorial approach to approximations of physical systems is
most effective when the "pictures" themselves are/or represent mathematical objects such
as those derived from topological or Euclidean geometry (1). For example, the Lewis
electron dot structures universally employed by organic chemists to represent molecular
structure display the connectivity of the atoms in a molecule and the location of valence
electrons on (lone pairs) or between (shared pairs) atoms. The Lewis electron dot
structures have the properties of a mathematical or topological object, called a graph,
and displays the graphical relationships between points (atoms) and connections of the
atoms with one another with lines (bonds). Indeed, the symbols R, *R, I, *I, *P and P of
Sch. 2.1 can be represented as Lewis structures (topological objects or chemical graphs)
or as three dimension representations of the position of the atoms in space (Euclidean
objects or chemical structures). The organic chemist is very familiar with the topological
and Euclidian representation of the structure of organic molecules, i.e., Lewis structures.
Even though such structures are, strictly speaking, mathematic abstractions, to the
chemist they take on a reality that leads to powerful structure-function, structurereactivity and structure-property correlations. Understanding that traditional molecular
structures are actually abstract mathematical objects may allow the student to be more
accepting of the use of wavefunctions as an alternate description of organic molecules.
Is it reasonable to translate wavefunctions into pictures? An a priori justification
is as follows. Although wavefunctions are formulated completely in mathematical terms,
nevertheless the physical basis for construction of wavefunctions are typically based on
the classical systems of physics (electrons, nuclei) which are capable of being readily
visualized. Thus, from this standpoint much of quantum mechanics does not really
provide an entirely new way of investigating natural phenomena; in many respects
quantum mechanics represents an evolution of classical mechanics which introduces
wavefunctions and quantization. In this sense we can develop a “quantum” intuition that
is based on classical mechanics of the world around us. The classical mechanics will be
usefully employed to provide the pictures we will use to understand processes that occur
in the quantum mechanical world. There will be, however, some features of quantum
mechanics which could not at all be considered as an evolution of classical mechanical,
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such as the consequences of electron exchange (Pauli Principle) and the indeterminacy of
measuring precisely certain quantities (Uncertainty Principle).
We shall see that a wave function of the species involved in Sch. 2.1 (R, *R, I, *I,
*P, P) can all be viewed as consisting of three visualizable components or separate
wavefunctions: a wavefunction which describes the positions in space of the electrons
(the electronic configuration), a wavefunction which describes the position of the nuclei
in space (the nuclear configuration) and a wavefunction which describes the position of
electron spins in space (the spin configuration). In this chapter we shall employ
electronic orbitals to visualize electronic configuration wave functions, vibrating
masses connected by springs to visualize nuclear configuration wave functions, and
precessing vectors to visualize electron spin configuration wave functions. Each of
these visualizations can be associated with the mathematical paradigms of quantum
mechanics in a qualitative manner that will allow a useful interpretation and
understanding of a wide range of photochemical phenomena.
2.2
Molecular Wave Functions and Molecular Structure
Conventionally a complete molecular wavefunction is composed of a complicated
and abstruse mathematical function given the symbol Ψ (the Greek capital letter “psi”).
Quantum mechanics provides an understanding of molecular structure, molecular
energetics and molecular dynamics based on the use of molecular wave functions Ψ and
on mathematical computations which use the wavefunction Ψ as a base function.
According to the paradigm of quantum mechanics, if the mathematical form of Ψ is
known precisely for a given molecule, it is possible, in principle, not only to compute the
electronic, nuclear and spin configurations of a molecule, but also to compute the
average value of any experimental observable property (energy, dipole moment, nuclear
geometry, electon spin, transition probabilities, etc.) of a given state of a molecule for an
assumed set of initial conditions and interactions.
But where do wavefunctions come from? According to the laws of quantum
mechanics, a wave function Ψ produced by solving the famous Schrödinger wave
equation (Eq. 2.1)
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HΨ = EΨ
(2.1)
In words, Eq. 2.1 is a mathematical expression of fundamental laws of nature,
which are considered in the same sense as Newton’s laws. According to the laws of
quantum mechanics, in nature the only possible stable (time independent or stable) states
correspond to Ψ, the wave functions which are solutions of Eq. 2.1 and the only allowed
energies are E, which are solutions of Eq. 2.1. In the wave equation H is called the
“Hamiltonian” operator and corresponds to a mathematical “operator” (a mathematical
procedure that changes one function into another) for the possible energies of the
system. These energies may be the electronic energies of the molecule, the vibration
energies of the molecule or the spin energies of the molecules. The special properties of
Eq. 2.1 are that the “possible” or “allowed” (stable) wavefunctions, Ψ, have the
remarkable property that when the mathematical operator H is multiplied by Ψ the result
is E times Ψ. This extraordinary mathematical property is associated with the mechanics
of stationary waves, where only certain waves or vibrations are stable and each of the
stable waves has an associated energy, E. These stable waves provide a powerful
analogy to the wave functions Ψ corresponding to stable states which have an energy E.
In the mathematics of quantum mechanics, the energy E (in general there will be a unique
energy associated with a ground state, R, excited states, *R and reactive intermediates, I),
a particular solution of the wave equation, is called an eigenvalue (“proper” value or
solution in German) and the wavefunction Ψ is called an eigenfunction.
In the following discussion, students who are unfamiliar with the mathematics of
quantum mechanics might think of the complete molecular wavefunction Ψ as a
mathematical representation of the entire molecular structure, electronic, vibrational and
spin. For the molecular structural representations of R, *R, I and P in Sch. 2.1, there exist
corresponding wave functions Ψ(R), Ψ(*R), Ψ(I), and Ψ(P), respectively. A system
whose wavefunction is an eigenfunction of some operator which represents a measurable
property of a system is said to be in a (stable) eigenstate for the measured property. The
explicit mathematic form of Ψ will not see the light of day in this text, but we will appeal
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to pictorial representations or graphs of Ψ and we shall relate these representations the
appropriate features of the familiar molecular structures.
Although the details of quantum mechanical calculations are beyond the scope of
this text, it is important to know that well established computational methods exist to
determine the wave functions of electron, nuclear and spin configurations of organic
molecules. The student is probably familiar with “stable waves” of macroscopic objects
such as vibrating springs, oscillating strings and tuning forks. These can be described in
terms of wave equations for which the eigenvalues correspond to stable vibrations or
oscillations.
Because of the mathematical complexity of the "true" or "exact" molecular wave
function Ψ, a means of developing approximate wave functions in employed in all actual
quantum mechanical computations. Fortunately, through the useful, if approximate and
qualitative use of quantum mechanics, it is possible to approximate the wave function and
translate it into a visualization that captures the essential features of the mathematics
underlying the wave function. Within this context of approximation, a picture, especially
a mathematical picture or geometry, may be a useful approximation if it contains rich
intuitive information and is capable of intellectual manipulation to make predictions and
direct interpretations of experimental results.
In addition to providing allowed wavefunctions Ψ to describe molecular systems
and to predicting the energies E of the states associated with the allowed wavefunctions,
quantum mechanics is concerned with predicting the allowed values of any molecular
property of interest. For example, one might ask the following questions such as the
concerning the species in Sch. 2.1:
(1)
What are the detailed electronic structure of *R and I?
(2)
What are the electron distributions of *R and I?
(3)
What is the probability of light absorption by R and the probability of light
emission by *R?
(4)
What are the rates of the photochemical and photophysical transitions of *R?
Quantum mechanics provide mathematical recipes for computation of any
experimentally measurable property, base on the mathematical manipulation of the
wavefunction. According to the laws of quantum mechanics, all the measurable
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information about a system is contained in the wavefunction Ψ. More generally than
computation of the energy, E, according to quantum mechanics, each observable
(measurable) property, of a system can be represented by an operator (we’ll use the
symbol P for a general operator and the symbol P for the property), a mathematical
procedure on the wavefunction of the system. The outcome of an observation of a
particular property is calculable by performing the appropriate mathematical operation
on the wavefunction Ψ (e.g., operation of the Hamiltonian operator, H, on Ψ produces the
allowed energies). The operator is related to some measurable property of a system
(energy, dipole moment, bond angle, angular momentum, transition probability, etc.) and
generally has a form similar to the mathematics that are used to compute the classical
property.
We finish this section with two important general conclusions derived from the
laws of quantum mechanics:
(1)
The only possible values of a measurement must be eigenvalues of equations of a
form similar to Eq. 2.2. For example, in general for every measurable property of a
molecular system there will be an operator P which operates on Ψ to produce
eigenvalues P. which correspond to an experimental measurement of that property of the
system.
PΨ = PΨ
(2)
(2.2)
Eq. 2.2 refers to a single measurement in an experiment on a single molecule. A
large number of experiments are actually performed on a huge number of molecules in an
real experiment so that an average value of the property, PAve is obtained in a laboratory
experiment. According to the laws of quantum mechanics, the average measurement,
PAve is obtained by integration of the form of Eq. 2.3 (a short hand for the actual
mathematics which involve complex functions and “normalization”).
P(observable property) = ∫ΨPΨ
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We shall now describe the most common approximation for Ψ that is employed to
solve Eq. 2.1 and thereby produce a set of allowed approximate wavefunctions and
approximate energies of a system. We then will use these approximate wavefunctions to
visualize the electron configurations, the nuclear configurations and spin configurations
of R, *R, I and P of Sch. 2.1. Next, we will describe some how to use these approximate
wavefunctions to estimate some properties of R, *R, I and P through the use of Eq. 2.3.
In these cases, in addition to visualizing the approximate wavefunctions, we will need to
identify and visualize the operator, P, corresponding to the property of interest and then
visualize Eq. 2.3 in which the appropriate operator is mathematically “sandwiched”
between the approximate wavefunction.
2.3 The Born-Oppenheimer Approximation. A Starting Point for Approximate
Molecular Wavefunctions
For organic molecules, the most important method for approximating molecular
wave functions was proposed and developed by Born and Oppenheimer.1 This
approximation states that since the motions of electrons in orbitals are much more rapid
than nuclear vibrational motions, electronic and nuclear motions could be treated
separately mathematically, in terms of approximate wave functions that describe the
electronic distribution in space and the vibrating nuclei independently. This
approximation greatly simplifies the solution of Eq. 2.1. The solutions of Eq. 2.1 under
the Born-Oppenheimer approximation provides approximate wavefunctions and
approximate energies. For singlet states, which are of main interest for organic
molecules (R and P) in their ground states the net spin is zero and is not considered in the
solution of Eq. 2.1. We have seen in Chapter 1, that *R and which are typically “open
electronic shell species” can be either singlet states or triplet state. However, since
electron "spin" motion is due to a magnetic interaction, and since magnetic and electronic
interactions interact only weakly for most organic molecules, the spin motion of electrons
may also be treated independently from both the motion of electrons in orbitals and the
motion of nuclei in space. The Born-Oppenheimer approximation is usually made when
wave functions are computed, and is generally an excellent approximation for a stable
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molecular state, such as the ground state of organic molecules, that does not interact
significantly with other (electronic, vibrational or spin) states. We shall see in Chapter 3
the conditions under which this approximation breaks down.
We now assume that although there is a mathematical function Ψ that represents
the "true" molecular wave functions of a molecule (i.e., is an exact solution of Eq. 2.1), in
practice, since it is impossible to determine Ψ exactly approximations are necessary.
Thus, we need to generate an approximate “Ψ” and furthermore, we shall seek to express
this approximate wavefunction in terms of three independent wave functions Ψ0, χ, S
which are the wavefunctions for the electron configuration, the nuclear configuration and
the spin configuration, respectively. The three approximate wavefunctions Ψ0, χ, S are
determined by solving Eq. 2.1 under the assumptions of the Born-Oppenheimer
approximation assuming a fixed nuclear configuration and no spin considerations and
then solving for the approximate “Ψ” and E. The relationship between Ψ, “Ψ” and Ψ0,
χ, S are given in Eq. 2.4.
Ψ
~
“Ψ”
~
"true" molecular wave function approx. molecular wave function
(Exact solution to Eq. 2.1)
(Solution to BO approximation)
Ψ0 χ S
(2.4)
orbital nuclear spin
(approximate wave functions)
The wave function Ψ0 represents an approximate electronic wavefunction (the
subscript zero refers to a “Zero Order” or initial working approximation) for electron
position and orbital motion in space about the positively charged (positionally frozen)
nuclear framework. The wave function χ represents the approximate vibrational wave
function (which will be discussed in detail in Section 2.18) for the fixed nuclear
framework. The wave function S represents the approximate wave function for spin
configuration (which will be discussed in detail in Section 2.21). Thus in Zero order, the
"true" molecular wave function, Ψ, is approximated by “Ψ” which in turn may be
approximated in terms of three separate, approximate wave functions, Ψ0 , χ , and S.
This approximation breaks down whenever there is a significant interaction between the
electrons and the vibrations (termed vibronic interactions) or between the spins and the
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orbiting electrons (termed spin orbit coupling). In essence it is the Born-Oppenheimer
approximation which justifies the visualization of Ψ in terms of the approximate wave
functions, Ψ0 , χ , and S and which makes the Lewis structures valid. We now proceed
to describe and visualize each of the three approximate wavefunctions in more detail.
First, let us consider the approximate electronic wave function, Ψ0, which we
interpret in terms of a picture of electrons in orbitals about nuclei. The detailed nature of
Ψ0 will depend on the level of sophistication desired. A common approach to
approximate is not to solve Eq. 2.1 for all the electrons of a molecule, but to solve for a
“one electron” molecule. The wavefunctions generated by this procedure are termed
“one electron orbitals”, ψi (given the symbol lower case “psi”) each of which possess an
eigen value of energy, Ei. For many qualitative analyses of molecular phenomena, Ψ0 is
usefully approximated as a composite of "one electron" molecular orbitals as shown in
Eq. 2.5,
Ψ0 ~ ψ1ψ2...ψn =
Πψi
(2.5)
where ψi is a solution of the wave equation for a "one electron" molecule. For a
discussion of these approximations, the reader is referred to any one of a number of
elementary texts.2 A fictitious molecule possessing only one electron does not experience
any electron-electron repulsions, clearly an approximation to a real molecule, but, as
experience has demonstrated, is still a remarkably useful approximation. For our
purposes in this text, it is only necessary to point out that the one electron wave
function ψ is at the level of approximation for orbitals that is conventionally given in
introductory texts or organic chemistry (the so-called Hückel orbitals). These are the
familiar orbitals that the student of organic chemistry has studied and so they will provide
a certain comfort level and intuitive appeal for students who are unfamiliar with quantum
mechanics. On the other hand, the wave functions corresponding to nuclear (vibrational)
configuration (χ) and electronic spin configuration (S) are generally not covered in
introductory chemistry courses and texts and therefore will be unfamiliar to the student.
As a result, we will develop pictorial representations of the wave functions χ and S in
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detail in this chapter in order to provide the student with some “quantum intuition”
concerning vibrational wavefunctions of spin wavefunctions and thereby how to visualize
vibrating molecules and electron spins according to the laws of quantum mechanics..
2.4
Important Qualitative Characteristics of the Approximate Wave Functions
The following qualitative characteristics of the approximate molecular wave
functions Ψ0 , χ , and S are of considerable general importance for an understanding to
the paradigm of Sch. 2.1:
(a)
the properties of a wave function (Ψ0 , χ , and S) are not subject to direct
experimental observation, but the properties of the square of the wave function
(Ψ02 (and Ψi2), χ2 and S2) is subject to direct experimental observation;
(b)
the square of a wave function (Ψ02 (and Ψi2), χ2 and S2) relates to the probability
of finding the electrons, nuclei, and spins, respectively, at particular points in
space in a molecular structure and therefore provide a means of pictorially
representing electrons, nuclear geometries and spins as geometric structures that
represent a molecule;
(c)
the functions Ψ0, χ, and S may be visualized as structures in three dimensions
with respect to a given nuclear configuration;
(d)
for molecules possessing local or overall symmetry elements, Ψ0 (and Ψi), χ, and
S will often possess useful symmetry properties that can be related to the motion
and spatial position of the electrons, nuclei, and spin, which will provide a
pictorial basis for the selection rules governing transitions between states.
In this chapter we shall be concerned with characteristics a, b and c above and we
shall see how qualitative knowledge of Ψ0, χ, and S can lead to a pictorial description of
the key electronically excited structures,*R(S1)and *R(T1), in the important working
paradigms of organic photochemistry (e.g., Sch. 1.3). In particular, we seek to
qualitatively estimate two important equilibrium (static) properties of molecular states:
(a) the various state electronic, nuclear, and spin configurations and (b) a qualitative
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ranking of the energies corresponding to the various state electronic, nuclear and spin
configurations. From knowledge of (a) and (b) we shall show how to readily construct
energy diagrams (Sch. 1.4) which allow the energetic ranking of the important low
energy states of a given molecule. In Chapter 3 we shall describe how to extend our
procedure to visualize operators which correspond to forces than trigger transitions
between states of a state energy diagram, all of which have a very similar nuclear
configuration (photophysical processes, *R → R). In chapter 6 we shall consider how to
use approximate wave functions to provide a pictorial description of transitions which
cause chemical changes in the nuclear configuration (primary photochemical processes,
*R →I and *R → F → P).
One may ask at this point, why deal with a wave function, such as Ψ, when it is
Ψ2 that is relevant in experiments. The short answer is that it turns out to be more
convenient to express the mathematical laws of quantum mechanics in terms Ψ rather
than in terms of Ψ2, yet nature has decided to connect measurable properties of molecular
systems to Ψ2 rather than to Ψ.
2.5
Postulates of Quantum Mechanics. From Postulates to Molecular Structure.
The Matrix Element.
As mentioned in the previous section, in this chapter we are concerned with how
knowledge of an approximate wave function can lead to a qualitative estimation of two
important equilibrium properties of molecular states: (a) the state energy and (b) the state
electronic, nuclear and spin configurations. From knowledge of (a) and (b) we can
construct state energy diagrams. The knowledge of the molecular properties shall come
from pictorial representations of the wave function.
The laws of quantum mechanics ar based on several postulates. These postulates
and the mathematics of wave functions for the basis of quantum wave mechanics (which
we shall term quantum mechanics for simplicity). One of the most important postulates of
quantum mechanics is that the average value, P, of any observable molecular property
(such as the energy of a state, the dipole moment of a state, the transition probability
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between two states, the magnetic moment associated with an electron spin, etc.) can be
evaluated mathematically in terms of a so called matrix element (Eq. 2.6),
Observable Property
P = <Ψ/P/Ψ>
(Experiment)
Matrix Element
(2.6)
(computation)
where P is a mathematical operator representing the forces or interactions operating on
Ψ to produce the property, P. When the property of interest, P, is the energy, E, of the
system, P is symbolized by H and is called the "Hamiltonian" operator or energy
operator. The quantitative evaluation of matrix elements is an important procedure in
theoretical chemistry, but the mathematical details of computation of matrix elements are
beyond the scope of this text.
The matrix element of Eq. 2.6 may corresponds to a measurable property such as
an energy or structural feature of a molecule (dipole moment, bond length, bond angle) of
a molecule or some dynamic feature of a transition (e.g., photophysical or photochemical
transitions of *R). For our purposes, the matrix element is the quantum mechanical
representation of an observable property of a molecular system. Instead of computing
the matrix element mathematically, we shall attempt to “qualitatively evaluate” the
matrix element by devising methods to visualize the components of the matrix element
(i.e., the wave function, Ψ and the operator, P) by attributing to them concrete structural
properties that can be associated with classical mechanics, which elegantly provides us
with pictorial representations, and thereafter by seeking qualitative conclusions based on
these admittedly oversimplified and approximate but useful models.
We can now use the approximate wave functions to compute matrix elements.
Thus, by combining Eq. 2.4 and Eq. 2.3, the approximate value of P is given by the
matrix element of Eq. 2.7.
P ~ <Ψ0χS/P/Ψ0χS>
(2.7)
In order to "understand" the factors determining the magnitude of P and to be able to
qualitatively evaluate its magnitude, we need to visualize the wave functions Ψ0, χ, S,
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and the operator P and then to perform qualitative “mathematical” operations on the
picture. From Eqs. 2.5 and 2.7, the approximate value of P is given by the matrix
element of Eq. 2.8.
P ~ <(Ψ1Ψ2...Ψn )χS/ P /(Ψ1Ψ2...Ψn )χS)>
(2.8)
We term this level of approximation of P as “working” or Zeroth-Order
approximation. By this we understand that we are within the Born-Oppenheimer
approximation (separation of electronic, nuclear, and spin motion) and that we are
dealing with one electron orbital wave functions (Hückel theory). In the next highest
level of approximation, the so-called, First Order approximation, we challenge some
aspect of the Zeroth-Order approximation and note its effect on the magnitude of P.
For example, we might begin to “mix” the one electron wavefunctions. Mixing wave
functions is nothing more that a mathematical processes which leads to a better
approximation of the wave function in First Order than we had in Zero Order. The new
wave function, if the mixing is done properly, is closer to the “true” wave function, Ψ. In
this approximation we only use the one electron wavefunctions that come from solution
of the simple one electron molecule model. For example, we can let the electron's orbital
motion mix with its magnetic spin motion (spin-orbit coupling), or we might introduce
the idea of electron-electron repulsion (electron correlation or configuration interaction),
or we might consider how vibrations mix electronic states (vibronic interactions), and
note how the magnitude of P is influenced as a result of these "First Order Mixing"
effects. If we have selected a good Zero Order approximation, the extent of mixing will
be small and the mixing can be handled by “perturbation theory” which will be discussed
below. If the mixing is strong, this is a signal that the Zero Order approximation is not a
good one and should be replaced by a better starting point to describe the approximate
wavefunctions.
In addition to making a First Order adjustment to the wave function, we may
modify the operator, P. For example, the Zero Order hamiltonian does not have electronelectron interactions or spin-orbit interactions included in it.
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2.6
The Spirit of the Use of Quantum Mechanical Wave Functions, Operators and
Matrix Elements
In the previous sections the wave function of a molecule, Ψ, was defined in terms
of a mathematical solution of a wave equation, Eq. 2.1. The spirit of the meaning of the
"true molecular wave function" Ψ is similar to the spirit of the great conservation laws
(energy, momentum, charge, mass), which we assume to be correct in detail, if we knew
all the details (the devil is in the details!). Similarly, if Ψ were known exactly and the
correct operators P for all of the observable properties of a molecule were know, then
according to the postulates of quantum mechanics Eq. 2.6 could be used to compute the
matrix element corresponding to values of all of the observable properties of a system
(state) Quite often a good level of approximation of the quantum mechanical system for
qualitative studies is achieved by appealing to classical mechanics at a starting point for
describing the system and then adjusting the system to include quantum effects. In these
cases approximate matrix elements are computed (Eq. 2.7 and Eq. 2.8)
Classical mechanics deals with observables such as position and momentum as
"functions" and postulates that all the information required to describe a mechanical
system is available if knowledge of the ”functions” describing the position and
momentum of the entities that make up the system is known. Newton's laws of motion
enable these "functions" to be displayed in a mathematical form.
In a similar fashion, quantum mechanics postulates that all the information about
a molecular system is contained in its wave function Ψ, and that in order to extract the
information about the value of an observable, some mathematical operation, P, must be
performed on the function (Eq. 2.3). This is analogous to the necessity of doing an act
(an experiment) on the system in order to make a measurement of its state. Problems in
quantum mechanics often boil down to (1) selecting the proper approximation to solve
for Ψ and/or (2) making the correct selection of the operator (P). Thus, the selection of
the correct "operators" (mathematical operations) or interaction leading to observable
properties is as crucial to the explicit solution of a problem in quantum mechanics as is
knowledge of the form of Ψ.
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Two of the key "operators" of quantum mechanics are related to the classical,
dynamical quantities of the momentum and position of a particles (electrons, nuclei and
spins) which are also critical in Newton’s equations. These operators allow the
determination of the energy levels of the system for the electrons, vibrations and spins.
Once the operators for these dynamical variables have been selected, it usually turns out
that operators for many observables can be set up in terms of these two fundamental
variables. In the same manner that Zero Order wave functions are employed to
approximate Ψ, in a quantitative evaluation of a matrix element, we almost always begin
with a workable Zero Order approximate Hamiltonian or energy operator, H0.which is
postulated to be the dominant set of interactions Ψ responsible for the energies of the
electronic orbital of the molecule. In the First Order approximation we consider
interactions, Hi that are "weak" (e.g., interactions of electrons with vibration and of
electrons with spins) relative to those described by H0 but which can be important in
causing transitions between the Zero Order states, especially Zero order states that are
close in energy.
In the text we employ the paradigms of molecular photochemistry to identify and
visualize the wave function operators which are plausible in determining the pathways of
photochemical reactions. These visualizations will provide us with the “quantum tuition”
that we seek to employ the power of quantum mechanics to the assist in the
understanding of photochemistry. The combination of visualization of the wave functions
and the interactions that operate on these wave functions will provide an intellectually
gratifying and practical, if approximate, understanding of most of the important features
of modern molecular organic photochemistry. Of course, we admit that there are pitfalls
and risks associated with such an approach, but we will accept it as a partial, if somewhat
flawed, understanding that is better than incomprehension, reminiscent of a life in
wonderland.
2.7
Atomic Orbitals, Molecular Orbitals, Electronic Configuration, and Electronic
States
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We propose to visualize a representation of the electronic portion of “exact”
wavefunction Ψ by approximating its electronic part, Ψ0 as a structure, namely a
configuration of overlapping, but non-interacting occupied one electron orbitals (Eq. 2.5).
It is assumed that the reader is familiar with the manner in which a molecular orbital is
approximated as a sum (or superposition) of atomic orbitals, Σψι, (e.g., Eq. 2.5).2 A
primary goal of this chapter is to develop a protocol for generating a state energy diagram
(Sch. 1.3) for a given stable ground state nuclear geometry of an organic molecule (R)
and for the low lying electronically excited states (*R) of the molecule employing one
electron orbitals.
The general procedure is as follows. For a given nuclear geometry (BornOppenheimer approximation) of a molecule, it is assumed that appropriate one-electron
(Hückel) molecular orbitals (MO')s are filled with the available electrons to generate a set
of molecular electronic configurations. Only the lowest energy configuration (ground
configuration, R = S0) and the first or second low-energy excited configuration (*R = S2,
S1 T1 or T2) generally need to be considered explicitly for the overwhelming number of
organic photochemical reactions. This situation corresponds to a Zero Order molecular
electronic configuration (Born-Oppenheimer approximation, one-electron orbitals). In
order to generate proper molecular electronic states we would have to take into account
electron-correlation in some manner in the First Order approximation which accounts
in some manner for electron-electron interactions which are missing in the one
electron molecule (only one electron, no electron-electron interactions!).
2.8
Ground and Excited Electronic Configurations
As an example of the procedure for constructing electronic configuration for the
ground state, we select the formaldehyde molecule H2CO as an exemplar. This exemplar
captures the important features of excited state configurations of many organic molecules
of photochemical interest and the lessons learned from the exemplar can be readily
extended to more complicated systems. The energies of the one electron MO's for H2CO
increase in the order shown in Eq. 2.9. In Eq. 2.9 1sO, 2sO, and 1sC refer to the
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essentially atomic, non-bonding MO's localized on oxygen (O) and carbon (C), and the
other MO's refer to conventional representations of bonding and antibonding orbitals.
1sO < 1sC < 2s0 < σCH< σCO < πC=O <nO<πC=O*<σCO*<σCH*
(2.9)
All of the bonding and non-bonding orbitals are filled in the ground state, S0.
From Eq. 2.9 we note that for S0 there exist a series of unoccupied antibonding orbitals,
the lowest energy of which is the π*CO orbital. In addition, unoccupied σ∗CH and σ∗CO
orbitals lie at very energies. The highest energy occupied molecular orbital in the ground
state of H2CO is the nO orbital. This orbital is termed the HOMO (or more compactly,
HO) orbital. The lowest energy unoccupied orbital of H2CO is the π*CO orbital. This
orbital is termed the LUMO (or more compactly, LU) orbital. We shall see that a simple
and useful approximation to the process R + hν → *R is to consider the absorption of
light as resulting from a transition between molecular orbitals (such as a HOMO
→ LUMO transition) that is induced by the absorption of a photon. We shall see that of
all of the MOs, the HOMO and LUMO generally are of greatest importance in organic
photochemistry. The HOMO and LUMO play a critical role in analyses of
photochemical processes, since these two orbitals determine the lowest energy transition
from R to *R. The initial photophysical and photochemical aspects start from *R, which
we shall see can be characterized as having one electron in the HO and one electron is the
LU. Now we address the issue of how to picture and describe the occupied orbitals in S0,
S1 and T1, the key excited states of the state energy diagram (Sch. 1.3).
What is the electronic configuration of the lowest energy electronic state of an
organic molecule, S0, for our exemplar molecule H2CO? H2CO possess 16 electrons
which must be distributed to the available MOs given in Eq. 2.9. In general, the energetic
order of the molecular orbitals, the number of available electrons, and the Pauli and
Aufbau principles are employed to determine the ground-electronic configuration of a
molecule. The ground state, S0, is constructed by adding electrons two at a time (Pauli
Principle) to the lowest energy orbitals (Aufbau Principle) first until all of the electrons
have been assigned to available orbitals. The configuration of the ground state of H2CO
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constructed according to these rules is represented by Eq. 2.10, in which the superscripts
designate the number of electrons in each orbital, the label in the () designates the kind of
orbital that is occupied and the subscript designates the nature of the orbital. The very
σCO* and σCH* orbitals are omitted for simplicity, since they are of such high energy
relative to the πCO*.
Ψ0(CH2=O) = (1s0)2(1sC)2(2sO)2(σCH)2(σ'CH)2(σCO)2(πCO)2(nO)2(πCO*)0
(2.10)
The total electronic distribution of an electronic configuration may be
approximated as a superposition of each of the occupied MO's that make up the
configuration (Eq. 2.5). However, as in the case of writing Lewis structures, it is a useful
and valid simplification to explicitly consider only the valence electrons when
considering chemical or photochemical processes, and considering the “core” electrons as
being close to the nuclei and too stable to be perturbed during chemical or photochemical
processes. In a further simplification, it usually suffices to consider explicitly, only the
highest energy valence electrons. In the case of CH2=O, we shall consider both πco and
no because they are of comparable energy and turn out to both be potential HOMOs in
substituted carbonyl compounds. We shall also indicate the πCO*orbital explicitly, even
thought it is unoccupied in S0, since the πCO* orbital is clearly the LUMO for CH2=O
and most carbonyl compounds. With the latter approximations in mind, a shorthand
notation for the ground state electronic configuration of CH2=O is given by Eq. 2.11:
Ψ0(CH2=O) = K(πC=O)2(no)2(πC=O*)0
Ground state, S0
(2.11)
In Eq. 2.11, K represents the "core" electrons (electrons in σ or lower energy
orbitals) that are "tightly bound" to the molecular framework, i.e., are close to, and are
stabilized by, the positive nuclear charge. For nearly all qualitative purposes to be
discussed in this text, the electrons symbolized by K are not significantly "perturbable"
during photophysical and photochemical processes. This “non-perturbability” is the basis
of the validity of the approximation given by Eq. 2.11.
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By an analogous protocol, if we ignore the lower energy MO's, we may describe
the ground state electronic configuration of ethylene (CH2=CH2) by Eq. 2.12:
Ψ0(CH2=CH2) = K(πC=C)2(πC=C*)0 Ground state, S0
(2.12)
In the case of CH2=CH2 it is sufficient to consider only πcc and πc=c* explicitly because
the electrons in the σ orbitals are of much lower energy and the σ* orbitals are of very
high energy. For CH2=CH2 the πc=c orbital is the HOMO and the πc=c* orbital is the
LUMO.
We are now in a position to describe the electronically excited states *R in terms
of orbital configuration. The lowest energy electronic states, which are most important in
organic photochemistry (see the energy diagram Sch. 1.3) possess an electronic
configuration for which an electron has been removed from the HOMO of the ground
state configuration and placed into the LUMO of the ground state configuration. As
examples, the lowest excited states of formaldehyde and ethylene possess the following
electronic configurations (Eqs. 2.13 and 2.14). The wave function for an electronically
excited state Ψ* is labeled with a * (termed "star") to emphasize that the state is
electronically excited.
Ψ*(CH2=O) = K(πC=O)2(no)1(π∗C=O)1
Electronically excited state, *R (2.13)
Ψ*(CH2=CH2) = K(πC=C)1(π∗C=C)1
Electronically excited state, *R (2.14)
The use of electron orbital configurations allows us to enumerate, to visualize, to
energetically rank and to conveniently classify electronic states, *R, and to describe
transitions between electronic states in electronic structural terms. For example, we
expect that formaldehyde (Figure 2.1a) will have two relatively low-energy electronic
transitions (n →π* and →π*) that will produce two corresponding electronic excited
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state configurations, *R(n,π*) and *R(π,π*). On the other hand, ethylene will have only
one low-energy electronic transition (π→ π*) and one corresponding electronic excited
state configuration *R(π,π*).
Figure 2.1
Schematic representation of (a) electronic transitions (b) electronic
configurations and (c) electronic states for the lowest excited states of formaldehyde
(top) and ethylene (bottom). The arrows for the states refer to the relative orientation
of the two electrons in the HO and LU.
In the above discussion we have developed an abbreviation Scheme for describing
orbitals and states that shall be employed throughout the text and which is widely
employed in the photochemical literature. In general, we shall describe ground
configuration of molecules in terms of the highest energy filled MO, or when appropriate
and useful, in terms of the two highest energy filled MO's. Electronically excited
configurations shall be described in terms of the two singly occupied MO's, usually the
HOMO orbital and the LUMO orbital (e.g., n,π* and π,π*). Transitions shall be
described only in terms of the orbitals undergoing a change in electronic occupancy (e.g.,
n  π* and π π*). This very handy short hand notation for electron configurations
also provides a useful method to represent excited states, such as S1 and T1 as we shall
show in the following Sections.
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2.9
The Construction of Electronic States from Electronic Configurations
The goal of this chapter was the development of ideas of molecular structure to
enumerate, classify, energetically rank and qualitatively visualize the electronic,
vibrational and spin nature of molecular states that appear in the state energy diagram of
an organic molecule (Sch. 1.4). We now make an important distinction between the
structure of singlet and triplet states that may be derived from the same electronic orbital
and nuclear geometric configuration when the same two orbitals are half-occupied, i.e.,
n,π* and π,π* configurations. The structural difference for the same orbital occupancy
for a given nuclear configuration resides in differences in the configuration of the spins of
the electrons in the two half filled orbitals and the operation of the Pauli principle.
2.10
Construction of Singlet and Triplet States from Electronically Excited
Configurations and the Pauli Principle3
The Pauli principle demands that any ground state configuration (such as that for
formaldehyde in which the electrons are all paired in orbitals) must be a ground state
singlet (in Section 2.26 we shall discuss the origin of the terms singlet and triplet), i.e.,
the spins of the two electrons which occupy a single orbital must be paired. In the
excited state two electrons are orbitally unpaired, i.e., each electron is in a different
orbital, one in a HOMO and the other in a LUMO. The Pauli principle allows, but does
not require, the spins of two electrons to be paired, if they do not occupy the same orbital.
As a result, either a singlet excited state in which there is no net spin (i.e., the two
electrons which are orbitally unpaired (HOMO and LUMO singly occupied) and electron
spins are antiparallel (↑↓) or paired as in the ground state) or a triplet excited state (the
two electrons which are orbitally unpaired (HOMO and LUMO singly occupied) and the
electron spins are parallel (↑↑) or unpaired) may result from the same electronic
configuration of two electrons in half-occupied orbitals. This means that each of the
same excited electronic configurations given in Figure 2.1b can refer to either a singlet
(S) or to a triplet (T) electronic state.
Thus, four states, two singlets and two triplets result from the two lowest energy
electronic excited electronic configurations of formaldehyde. The lower energy n,π*
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state may be either a singlet (S1) or a triplet (T1) configuration, and similarly the higher
energy π,π* state may be either a singlet (S2) or triplet (T2) configuration. These states are
displayed in Fig. 2.1c (See a more detailed description of the orbital configuration and
spin occupancy in Fig. 2.2).
2.11
Characteristic Configurations of Singlet and Triplet States: A Shorthand
Notation
The conventions described in Chapter 1 for the state energy diagram of a
molecule (Sch. 1.4) serve as a basis for a convenient notation to describe electronic
configurations and electronic states. We term the ground (lowest energy) electronic
singlet state S0, where S indicates there is no net electronic spin associated with the state
(i.e., each spin up is matched by a spin down, ↑↓) and the subscript zero refers to the
ground electronic state. We reserve the subscript zero for the ground electronic ground
state S0. Therefore, for triplet states with parallel spin (↑↑) we label the first triplet level
located energetically above S0 as T1, where T indicates a threefold degeneracy of the
state (in a magnetic field) due to the three possible alignments of two unpaired spins (to
be discussed below in Section 2.27) and the subscript 1 refers to the energy ranking
among triplet states relative to S0.
We label electronically excited singlet states S1, S2, etc., where the subscript
refers to the energy ranking of the state relative to the ground state (arbitrarily given the
rank of energy equal to zero and all other states possessing positive energy relative to the
ground state). Thus, S1 is the first electronically excited singlet state located
energetically above S0, and S2 is the second electronically excited singlet state located
energetically above S0. Similarly, T1 is the first electronic triplet state located above S0
and T2 is the second electronic triplet state above S0.
As a shorthand, we describe electronic and spin configurations in terms of the key
molecular orbitals that are expected to dominate the energy and/or chemistry of the
configuration. Thus, for formaldehyde we need to explicitly consider only the πC=O, nO,
and π*C=O orbitals in discussing electronic configurations, and hence in discussing
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electronic states, in a state energy diagram. This approximation assumes that
consideration of the lower lying σ orbital and the higher energy σ* orbital is not
necessary for an understanding of the photochemistry of formaldehyde. As a result of the
above discussion we may state the following rule: Each electronic state may be described
(a) in terms of a characteristic electronic configuration, which in turn may be often be
described in terms of the HOMO and LUMO orbitals: (b) in terms of a characteristic spin
configuration, either singlet (↑↓) or triplet (↑↑).
In summary, the use of electronic and spin configurations to describe the S0, S1,
S2, T1 and T2 states of formaldehyde may be described as shown in Table 1. In addition
based on the individual orbital energies, we can rank the energy of the n,π* state of a
given spin as unambiguously being lower in energy than the corresponding π,π*
states of the same spin, e.g., S1(n,π*)
Table 1. States, characteristic orbitals, characteristic spin configurations and
shorthand description of low lying states of formaldehyde.
State
Characteristic
Characteristic
Shorthand
Orbitals
Spin Electronic
Description
Configuration
of State
S2
π,π*
(π)1(n)2(π*)1
1(π,π*)
T2
π,π*
(π)1(n)2(π*)1
3(π,π*)
S1
n,π*
(π)2(n )1(π*)1
1(n,π*)
T1
n,π*
(π)2(n)1(π*)1
3(n.π*)
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S0
π,n
(π)2(n)2
π2n2
In many cases a single electron configuration is adequate to approximate the
electronic characteristics of an electronic state. In some cases, however, a combination
of two or more electronic configurations will be required to achieve a good
approximation of a state. This is the situation when two states in Zero Order have similar
energy and can be coupled by some interaction, say a vibrational motion. In this case,
we say the a single electron orbital are inadequate as a description because these orbitals
are "mixed", i.e., an n orbital may mix in some π character as the result of a vibration
(Section 3.6). We shall see examples of “mixed” n,π* and π,π* states, where neither
“pure” configuration provides an adequate representation of the actual state, but the
“mixed” or “First Order” representation works well. Perturbation theory teaches the
rules for such mixing and will be discussed in Chapter 3.
2.12 Electronic Energy Difference between Singlet and Triplet States
The student may recall from General Chemistry Hund's rule, which states that for
atoms a triplet state that is derived from a given electronic configuration possessing two
half-filled orbitals is always of lower energy than the corresponding singlet state derived
from the same configuration, i.e., 1E (n,π*) > 3E (n,π*) and 1E (π,π*) > 3E (π,π*). A
physical and theoretical basis for Hund's rule is available from consideration of the
implications of the Pauli principle. The Pauli principle operates as a sort of quantum
mechanical "traffic cop for electron motion" which "instructs" the two key orbitally
unpaired electrons in triplet states how to "avoid" one another. The Pauli principle
requires the electrons to correlate their motions because two electrons of the same spin
are absolutely not allowed to occupy the same space at the same time. The electrons in a
singlet state are not compelled to obey the Pauli principle and may on occasion approach
the same region of space. This will always serve to raise the energy of the singlet state
relative to the triplet. Thus, by being instructed to avoid being in the same place,
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electrons in a triplet state are able to minimize electron-electron repulsions relative to a
singlet state in which the electrons are allowed to approach without the Pauli restriction.
Thus, the difference in electronic energy, ΔEST, between singlet and triplet states
(that are derived from the same electron orbital configuration) results from the "better"
correlation of electron motions in a triplet state producing lower electron-electron
repulsions. Although Hund’s Rule strictly speaking refers to the states of atoms, it may be
applied to molecules also. In order to visualize how this singlet-triplet energy separation
(ΔEST) arises for a given orbital configuration of a molecule and to gain an appreciation
of how the magnitude of the ΔEST depends upon the orbitals comprising the
characteristic configuration, we attempt to visualize the magnitude of matrix elements
that evaluate the electronic energy of orbitals, configurations.
2.13
Evaluation of the Singlet-Triplet Energy Gap for Electronic States of the Same
Configuration
A state energy is the summation of all of the Zero Order energies (one-electron
orbital, no electron-electron interactions) plus electron-electron repulsion energies in First
Order. In the Born-Oppenheimer approximation, the nuclear geometry is fixed and the
attractive forces between the negative electrons and the fixed positive nuclear framework
are assumed to contribute a certain stabilization energy. The differences in energy
between different states in this approximation is due entirely to electron-electron
repulsions. Thus, when we discuss electronic energy differences between two states we
are referring to the differences between electron-electron repulsions in the two states.
The state with the smaller electron-electron repulsions has the lower energy. The classical
operator for the energy increase due to electron-electron repulsions is defined by the
familiar electrostatic or Coulombic repulsion between electrons, i.e., e2/r12 where e is the
charge of an electron and r12 is the separation between the electrons. The Coulombic
interactions between two electronic distributions is always repulsive and the value of this
electronic repulsion may be computed by integration of the repulsive interactions over all
of the volume of a molecule assuming a fixed positive molecular framework (BornOppenheimer approximation). The matrix element for the total electron-electron
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repulsion energies are broken down into two types: (1) the repulsions between electrons
due to classical mechanical electrostatic interactions (termed Coulombic interactions and
given the symbol K) and (2) the repulsions between electrons when quantum mechanical
interactions due to the Pauli Principle (termed electron exchange interactions and given
the symbol J) are taken into account. J is referred to as the electron exchange integral
because the Pauli Principle is derived from a postulate on the symmetry property of
electrons when two electrons exchange positions in space with one another. The
exchange integral is a purely quantum mechanism phenomena and amounts to a quantum
mechanical correction to the classical electronic distribution which does not take into
account the influence of electron spin on electron-electron correlation and therefore
electron-electron repulsion. Quantum intuition comes from the acceptance of the Pauli
Principle as follows. When the nuclei of two atoms come close enough together (ca 3 Å
or less) the wavefunctions (electron clouds) of the two atoms begin to overlap slightly.
Although the individual wavefunctions may not be significantly perturbed as they
approach, when even a small amount of overlap occurs, the electron from one atom will
have an unavoidable tendency to slip from the nucleus of its atom and visit the nucleus of
the other atom. This intuition when coupled with that Pauli concept that quantum spin
correlation which causes electrons of the opposite spin (↑↓) to have an enhanced
probability of being found near one another (in spite of the added repulsions that
proximity brings) and simultaneously causes electron of the same spin (↑↑) to stay apart
(and possess a lower energy than expected from classical considerations).
As an exemplar let us consider the energies, E, of the ground state and the lowest
excited S1 and T1 states of formaldehyde based on the energies of the n and π* orbitals
(Fig. 2.1). The energies of S0, S1 and T1 are defined by Eqs. 2.15, 2.16 and 2.17,
respectively.
E(S0) = 0
by definition
(2.15)
E(S1) = E0(n,π*) + K(n,π*) + J(n,π*)
(2.16)
E(T1) = E0(n,π*) + K(n,π*) - J(n,π*)
(2.17)
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In Eqs. 2.15 and 2.16 E0(n,π*) is the Zero Order energy of the one electron
excited state, K(n,π*) is the First Order Coulombic correction due to classical electronelectron correlation and J(n,π*) is the First Order quantum mechanical correction due to
the Pauli Principle. Since the charges are the same for electron-electron electrostatic
interactions which lead to repulsions, the values of both of the energy integrals K and J in
eqs. 2.12 and 2.13 are defined as mathematically positive (corresponding to repulsions).
The energy of the Coulomb integral K(n,π*) corresponds to the contribution of the
classical charge interactions of electrons (for a fixed framework of positive nuclei) to the
total energy of the molecule. As mentioned above, since the energy computed is that for
the Coulombic interactions between electrons and therefore between the same charges,
the value of K(n,π*) is always positive, corresponding to an energy increasing repulsive
interaction between electrons.
The exchange energy, J(n,π*), is the correction that must be made to the
computation of the total electronic energy of a molecular to take into account the effects
of the Pauli Principle, which requires a correlation of electrons which depends on the
orientation of their spins. A simple picture of the Pauli Principle is that quantum
mechanics requires electrons of the same spin (↑↑) avoid approaching one another, but
that electrons of opposite spin (↑↓) actually have an enhanced probability of being found
near one another. Thus, we build up the quantum mechanical “intuition” that if two
electrons have the same spin, the average repulsion energy will be less than the repulsion
computed from the classical model because of the tendency of electrons with parallel
spins to avoid each other; on the other hand, if two electrons have the same spin, the
average repulsion energy between them will be greater than the repulsion computed from
the classical model because of the tendency of two electrons of the same spin to couple to
one another. As a result, for the T1 state, the average repulsive energy is reduced from
the expected value for the classical case, i.e., E(T1) = E0(n,π*) + K(n,π*), to a lower
value E0(n,π*) + K(n,π*) - J(n,π*). We obtain the following “quantum intuition” from
the above discussion. The average repulsions between electrons are different for the
same orbital configuration for singlets and triplets because when electrons have the same
spin orientation (↑↓) they tend to “stick together” and suffer energy increasing repulsive
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interactions, whereas in the electrons have opposite spin orientation (↑↑) then inherently
avoid each other. Thus, for the same electronic configuration, the average energy due to
electron repulsions for a triplet state, T1 will always be lower than the energy of electron
repulsions for the corresponding singlet state, S1 of the same electron configuration.
Let us now estimate the difference in energy between a singlet and triplet state of
the same electronic configuration. Starting with the Zero Order approximation (one
electron orbitals) for a n,π* state, let E0(n,π*) be the matrix element for the Zero Order
one electron orbital energy. In this case the electrons are assumed not to interact, so that
the singlet and triplet energies would be identical, i.e., J and K = 0). Let the matrix
element J(n,π*) be the value of the electron repulsion when the electron exchange is
taken into account, and let K(n,π*) be the matrix element that measures ordinary classical
electron repulsion due to Coulombic interactions between electrons. We can obtain ΔEST,
the energy difference between S1(n,π*) and T1(n,π*) by subtraction of Eq. 2.17 from
2.16 to achieve Eq. 2.18, and that the values of E0(n,π*) and K(n,π*) drops out of the
evaluation so that ΔEST only depends on the value of 2J(n,π*)!
ΔEST = E(S1) - E(T1) = 2J(n,π*) > 0
(2.18)
Since J(n,π*) corresponds to electron-electron repulsion and must correspond to a
positive (repulsive) energy we conclude, within this approximation, that E(S1) > E(T1) in
general for all n,π* states! In addition, the value of ΔEST is precisely equal to 2J(n,π*),
i.e., the energy gap between the singlet and triplet equals twice the value of the electron
exchange energy. Since the details of the orbital were not considered in making the
argument, we conclude that for a π,π* electronic configuration, S(π,π*) and T(π,π*),
ΔEST = 2J(π,π*). This qualitative result can be generalized to the following conclusion:
the energy gap between a singlet state and a triplet state of the same configuration is
purely the result of electron exchange. It follows therefore that we can estimate the value
of ΔEST if we can estimate the value of J. Let us now seek to answer the question, which
is larger, in general, ΔEST(n,π*) or ΔEST(π,π*)? We shall answer this question by using
a pictorial approach.
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2.14
An Exemplar for the of Singlet-Triplet Splittings9
We now consider our first pictorial representation of electronic orbitals as a
means of qualitatively evaluating the value of a matrix element, namely the matrix
elements for the values of energy differences ΔEST(n,π*) and ΔEST(π,π*).9 The
estimation of ΔEST requires the estimation of the matrix elements, J(n,π*) and J(π,π*),
the electron exchange integral (Eqs. 2.16 and 2.17). We shall consider as a specific
exemplar, the (n,π*) and (π,π*) states of ketones. The important photophysics of ketones
is described in Chapters 4 and 5 and the photochemistry of ketones is described in
Chapter 9. The lowest S1 and T1 states of many ketones are n,π* states, with the S2(π,π*)
and T2(π,π*) being somewhat higher in energy, as is the case shown in Fig. 2.1. The
value of J(n,π*) represents the electrostatic repulsion, due to electron exchange, between
an electron in an n orbital and an electron in a π* orbital.
The magnitude of J(n,π*) is given by a matrix element shown in Eq. 2.19, in
which n and π* represent the wavefunctions for the n and π* orbitals, respectively. The
numbers refer to the electrons occupying these orbitals and e2/r12 is the operator (e is the
charge on an electron and r12 is the distance of separation between electrons) which
represents the repulsion between the exchanging electrons. The latter term may be
factored out of Eq. 2.19 so that the value of J(n,π*) can be seen to be proportional (Eq.
2.20) to the mathematical overlap of the wave functions of the n orbital with the wave
function of the π* orbital. The magnitude of this orbital overlap is represented by the
symbol <n|π*>, termed an overlap integral.
J(n,π*) = <n(1)π*(2)|e2/r12|n(2)π*(1)>
(2.19)
or
J(n,π*) ~ e2/r12<n(1)π*(2)|n(2)π*(1)> ~ <n|π*>
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In other words, the magnitude of J(n,π*) will be proportional (Eq. 2.20) to the
value of integral <n|π*>. The value of this integral is proportional to the degree of
overlap of the n and π* orbitals which we can estimate qualitatively pictorially. An
overlap integral, such as that of Eq. 2.20, may be pictured as a measure of the mutual
resemblance of two wavefunctions. If the two wavefunctions are identical, then the
normalized value of the overlap integral, <|> = 1. On the other hand, if the two
wavefunctions do not overlap at all then the normalize value of <|> = 0. We may
visualize the overlap integral <n|π*> if we replace the symbol n (which represents the
wavefunction of the n orbital of formaldehyde) with a picture of the n-orbital and the
symbol π* (which represents the wavefunction of the non-bonding nO orbital of a ketone)
with a picture of the π C=O* orbital (Eq. 2.21) and we picture the common regions of
overlap of these orbital in the space about the nuclei of formaldehyde. We then
qualitatively estimate the degree of overlap between the n and π* orbital. We note that
the overlap integral must be “small” because these orbitals do not occupy very much of
the same region of space. By inspection an electron in the π* orbital has its density above
and below the plane of the molecule (assuming a planar structure). The lobe of the π*
orbital on the carbon atom does not overlap at all with the n orbital (Eq. 2.21b).
Furthermore, the n orbital is in the nodal plane of the π* which means that at this level of
approximation there is exact canceling and zero overlap of the two orbitals!
Now let us compare this result to the overlap of orbitals for a π,π* configuration.
Employing the ideas leading to Eqs. 2.19 and 2.20 we obtain for a π,π* configuration
Eqs. 2.22a and 2.22b.
Jπ,π* ~ <π|π*>
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From a comparison of the orbitals we see that in this case there is significant overlap at
both the C and the O atoms. It is clear that Jπ,π* will be greater, in general, than Jn,π*
because the overlap integral <π|π*> will generally be greater than the overlap integral
<n|π*>.
We can now use a physical interpretation of the overlap integrals to understand
and compare the magnitude of the singlet triplet gaps for n,π* and π,π* states. The
overlap integrals of Eqs. 2.21 and 2.22 are measures of the region of space that is
common to two orbitals. If the value of the overlap integral is large than the region of
space occupied by the electrons in each orbital is large and the electron-electron repulsion
is large. If the value of the overlap integral is small, then the region of space occupied by
the electrons in each orbital is small and the electron-electron repulsion is small. The
energy “advantage” that that Pauli principle provides is that the triplet decreases as the
space available for electron repulsions decreases (as the e2/r12 term in Eq. 2.19 decreases
and the orbital overlap decreases).
From the above pictorial arguments we conclude
that in general ΔEST(n,π*) < ΔEST(π,π* ) because the overlap of a π with a π* orbital
will in general be greater than the overlap of a n and π* orbital.
Now we need to challenge this qualitative pictorial conclusion with some
quantitative data to see if it is valid. Experimentally, values of ΔEST(n,π*) states for
ketones are 7-10 kcal/mole, while values of ΔEST(π,π* )states for aromatic hydrocarbons
are of the order of 30-40 kcal/mole. Table 2.2 lists some experimental values of ΔEST
and shows that indeed states derived from n,π* configurations consistently have smaller
singlet-triplet splitting energies than states derived from π,π* configurations.
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Table 2.2. Some Examples of Singlet-Triplet Splittings
Molecule
Configuration S1 and T1
ΔEST (kcal/mole)
CH2=CH2
π,π*
70
π,π*
40
π,π*
35
π,π*
35
CH2=O
n,π*
10
(CH3)2C=O
n,π*
7
Ph2C=O
n,π*
7
The state energy level diagram for formaldehyde (Figure 2.1 c) may now be
modified to include the singlet triplet splittings (Figure 2.2). From this point on, state
energy level diagrams will be presented at least at this level of approximation.
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Figure 2.2
State energy diagram for formaldehyde including singlet-triplet splittings
and electronic configurations of states.
2.15
Summary
The solutions of the Schroedinger wave equation (Eq. 2.1) provide the
mathematical form of the wavefunctions (Ψ) of a system. Through the wavefunction, the
chemist has access to computation of all the measurable properties (P) of a system
through Eq. 2.3 and the matrix elements corresponding to Eq. 2.6. The electron
distribution of organic molecules can be described in terms of wavefunctions
representing electronic configurations composed of one electron orbitals, ψι. This
approximate description provides a qualitative framework for not only visualizing the
distribution of electrons in space, but also as a means of estimating the magnitude of
matrix elements for important quantities such as singlet-triplet energy gaps. We shall
rely heavily on the one electron orbital model in this text and show that it is useful not
only in visualizing and evaluating “static” properties such as the electron distributions,
but also in visualizing and evaluating “dynamic” properties such as the probabilities of
transitions between electronic states (Chapter 3).
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2.16
Vibrational Wavefunctions. The Classical Harmonic Oscillator
We have developed one electron wave functions (ψ), and configurations of one
electron orbitals (Πψi)as models for approximation and visualization of the electron
orbital portion of the approximate wave function Ψ0. Now we consider how to
approximate and visualize χ, the wavefunction of the vibrational portion of Ψ0.
Chemists are accustomed to considering vibrations in terms of nuclei oscillating back and
forth in the potential field of the electrons (the Born-Oppenheimer approximation) and
executing motions approximated by those of a classical harmonic oscillator. The
classical harmonic oscillator is, therefore, a useful starting point for visualizing χ.
Following the classical interpretation of the harmonic oscillator, the relationship of the
classical oscillator system to the quantum oscillator will be developed.
A few comments concerning the basic features of the harmonic oscillator are in
order. It is important to recognize that at the quantum level of the photon and the
molecule, many important quantum systems can be approximated as vibrating harmonic
oscillators: i.e., the vibrations of the electromagnetic field, the orbital motion of an
electron about a nucleus, the vibrations between two atoms of a chemical bond and the
precession of an electron spin about a coupling magnetic field. The importance of the
harmonic oscillator is found in the fact that the same mathematics can be employed to
treat an initial, approximate description of many physically distinct systems, all of
which respond to a distorting or perturbing force in the same way. In a sense when we
use the mathematics of harmonic motion we are discussing all these problems at once,
even as we focus on one selected exemplar.
A harmonic oscillator may be defined as any system that, when perturbed from its
equilibrium, experiences a restoring force, F, that is proportional to the displacement
from its equilibrium position, re, Eq.2.23,
F = -kr,
(2.23)
where r is the distance of displacement from equilibrium, F is the restoring force and k is
termed the proportionality constant related the force to the extent of displacement. The
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proportionality constant k is termed the force constant for oscillation. A harmonic
oscillator at rest may be set into oscillation by some impulse or perturbation that is
applied suddenly. The perturbation sets the system into “harmonic oscillations”. An
oscillating pendulum or a vibrating string of a violin or an oscillating tuning fork are
common examples of harmonic oscillators.
When Eq. 2.23 is applicable to a system of interest, from classical physics the
potential energy (the derivative of the force as a function of r) of the system can be
related to the displacement r by Eq. 2.24
PE = 1/2kr2
(2.24)
The most common vibrations of an organic molecule may be classified as
stretching vibrations between two bound atoms (e.g., C-H or C=O stretching) or bending
vibrations between three bound atoms (e.g. H-C-H or H-C-C bending). Essentially all of
the important features of the harmonic oscillator model for vibrations are incorporated
into the characteristics of the stretching motion of a diatomic molecule (to only vibration
available for such systems). Thus, we shall analyze a generalized diatomic molecule and
apply the general results for all vibrations of an organic molecule.
The harmonic oscillator (Eq. 2.24) is often a good approximation of a vibrating
system when small displacement from equilibrium positions of the atoms are considered.
Let us consider a vibrating (oscillating) diatomic molecule, X-Y. For a classical harmonic
oscillator, the relative vibrational motion of X with respect to Y is a periodic, oscillating
function of time. The frequency, ν, of oscillation follows Eq. 2.25, where k is the same
force constant (Eq. 2.23) of the chemical bond between the atoms X and Y and m is the
reduced mass of the two atoms forming the bond. The significant characteristics of a
classic harmonic oscillator are: (1) the amplitude (displacement) of vibration increases
with the total energy of the oscillator; (2) the frequency of vibration, ν, is determined
only by the mass, m, and force constant, k, and not by the amplitude of the motion; (3)
the larger the force constant, k, for a given pair of masses of reduced mass m, the higher
the frequency, ν; (4) the smaller the reduced mass, m, for a given force constant, k, the
higher the frequency, ν.
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ν
=
(k/m)1/2
(2.25)
According to Eq. 2.24, a plot of the potential energy (PE) of a vibrating diatomic
molecule as a function of internuclear separation, r, is given by the familiar classical
parabolic potential-energy curve (Fig. 2.3). At some particular internuclear separation
re the potential energy of the system is at a minimum, i.e., at the separation re the nuclei
possess their equilibrium configuration. If the separation is decreased to a value less than
re (to the left in the curves of Fig. 2.3) the potential energy of the system increases rapidly
as a result of internuclear and electronic repulsions. On the other hand, if the internuclear
separation is increased to a value greater than re (to the right in the curves of Fig. 2.3) the
potential energy also increases due to the stretching of the X-Y bond. At the equilibrium
position, re, there is no net restoring force operating on the atoms, but at any displacement
from re, a restoring force, F in Eq. 2.23 exists. The restoring force thus varies in
magnitude and in direction in a periodic fashion at frequency ν in equal intervals of times
called periods, τ. We see that a pair of atoms undergoing classical harmonic motion
stretch and contract back and forth through a point of separation re at which the potential
energy, PE, of the system is a minimum. Since we will generally be interested in energy
differences between vibrational levels, we may arbitrarily define the potential energy
minimum as Eo, with Eo defined as the zero of energy and all other vibrational energies
being positive (higher energy, less stable) relative to this value.
These features lead immediately to conclusions such as the expectation (1) that
C-H bonds will tend to be of high frequency, because they are possess large force
constants (strong bonds, like a stiff spring holding the atoms together) and possess a
small reduced mass (H is the lightest atom) and (2) that weak bonds between two heavy
atoms such as a C-Cl bond will be of very low frequency because they possess small
force constants (weak bonds, like a soft spring holding the two atoms together) and also
possess large reduced mass. Fig. 2.3 left shows an example of the shape of the potential
curve of a vibrating C-H bond and Fig. 2.3 right shows an example of the shape of the
potential curve for a C-Cl bond. The harmonic oscillator provide powerful “quantum
insights” to the properties of vibrating atoms. The stretching frequencies of C-H bonds
which combine a strong bond and a small reduced mass are of the order of ν ∼ 1014s-1.
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Such a frequency is among the highest found for organic molecules. This frequency
corresponds to a wavelength of ~ 3000 nm . On the other hand, the stretching frequency
of a C-Cl bond, which combines a relatively weak bond and a large reduced mass
possesses are relatively low in frequency 1013 s-1. This frequency corresponds to a
wavelength of ~ 30000 nm.
Sch. 2.2 schematically depicts how a classical vibration between atoms X and Y
can be visualized via a model in which an atom (mass) X is attached to an atom (mass) Y
by a spring (representing the bond between X and Y). X and Y vibrate back and forth
with respect to one another along the bond axis at the frequency, ν (Sch. 2.2a). If one of
the two atoms is much lighter than the other (e.g., HCl) nearly all of the motion in space
is due to the lighter mass (e.g., H in the case of HCl). In this case we can assume the
heavier mass to be essentially stationary. In effect, we can consider the heavy atom to
serve as a "wall" to which the lighter mass is attached by a spring (Sch. 2.2b).
(a)
(b)
(( X
X
Stretching
Y ))
Y ))
(( X
Y ))
X
Y ))
Contracting
Sch. 2.2. Schematic of stretching and contracting motions of a diatomic molecule. See
text for discussion.
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Figure 2.3. Potential energy curves for a classical diatomic molecule. Left: a diatomic
molecule with a strong bond and/or light atoms. Right: a diatomic molecule with a
weak bond and/or heavy atoms.
V = (1/2)k(Δr)2
(2.21)
At any point on the potential-energy curves of Figure 2.3, the vibrating atoms
experience a restoring force F, attracting the system towards the equilibrium geometry, re.
The magnitude of this force, F, is given by F = dPE/dr, i.e., the force is the ratio of the
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magnitude of the potential energy of the system at any point to the displacement from
equilibrium Δr = |r - re|. The value of the slope is a measure of the resistance of the
system to distortion from the equilibrium position. From Eq. 2.24 we see that for a strong
bond (large k) a large change in potential energy, PE, will occur with only a small
displacement because PE is a quadratic function of the displacement. A strong bond thus
corresponds to a potential curve with "steep" walls (Figure 2.3 left), such that even small
displacements cause a large increase in the potential energy. A weak bond corresponds
to a potential energy curve with "shallow" walls (Figure 2.3 right), such that small
displacements cause a large increase in potential energy. Since the frequency of
vibration, ν, is directly proportional to the bond strength and inversely proportional to the
mass, a steep potential curve implies a higher frequency vibration and a shallow potential
curve implies a lower frequency vibration. Thus, we expect the potential energy curve on
the left of Figure 2.3 to be characteristic of a strong bond and the he potential energy
curve on the right of Figure 2.3 to be characteristic of weak bonds, if we compare similar
reduced masses.
The term "representative point" on a potential energy curve may now be defined.
We refer to a particular point on the curves of Figure 2.3 as a "representative point"
corresponding to a specific displacement, Δr, on the x- axis, and which "follows" the
motion (kinetic energy, KE), displacement from the equilibrium separation (Δr), and
potential energy (PE) of the pair of atoms XY. For example, when the representative
point, r, is at re, the molecule possesses zero PE (by definition). It may or may not
possess KE at this position. For any other value of r on the PE curve, the molecule
possesses excess potential energy. The molecular structures corresponding to such points
are kinetically unstable, i.e., they are attracted toward re by restoring forces, just as a
classical particle on a downward-sloping surface is attracted toward a potential energy
minimum. Therefore, we expect any representative point at a position r to move
spontaneously toward re.
Consider the points a, b Fig. 2.3 left and c, d Fig. 2.3 right which correspond to an
excess of vibrational energy of the system and are the “turning points” of vibration.
Suppose the representative point is initially at the turning point, a (Fig. 2.3 left), in its
motion. The point will spontaneously be attracted to the equilibrium position because of
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the restoring force operating on it. In moving towards re the system picks up kinetic
energy. The total vibrational energy (Ev) of the system equals (Eq. 2.26) the potential
energy (PE) + the kinetic energy (KE). The latter is maximal when the point is at re
where the PE is zero. On the other hand, at the turning point, the KE is zero so that the
total energy is equal to the PE of the system. In the absence of friction, we can visualize
the representative point as oscillating back and forth between a and b. This oscillation is
represented by the arrows on the curves. Similar concepts apply for oscillations between
c and d. The latter occur at a lower frequency than those between a and b.
Eν = PE + KE
(2.26)
In summary, the classical motion of vibration of X relative to Y in a diatomic
molecule may be visualized as the motion of a representative point moving back and
forth from one position on the potential energy curve to another. For example, for the
same amount of excess vibrational energy, the vibrating atoms in a strong bond (Fig.
2.14, left) will experience a smaller displacement than the vibrating atoms in a weak
bond (Fig. 2.14, right) because of the steeper “walls” of the potential energy curve on the
left.
2.17
The Quantum Mechanical Version of the Harmonic Oscillator
With the classical harmonic oscillator of two vibrating masses as a model, we
now consider the quantum mechanical system for a vibrating diatomic molecule. As for
all quantum objects, the vibrating diatomic molecule is described by a vibrational wave
function, χ. For a given electronic state, the wave function χ describes the shape
(position in space relative to the electron cloud) and motion of the nuclei, just as Ψ
describes the shape (position in space relative to the nuclei) of the electrons. We have
discussed the orbital model for visualizing the electronic portion of a wave function.
How can we visualize χ, the vibrational wave function and how do impose quantum
mechanical effects on the classical harmonic oscillator?
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Solution of the analogue of the electronic wave equation (Eq. 2.1) for a harmonic
oscillator yields a set of vibrational wave functions χι each of which possesses a unique
potential energy PEv. Examination of the properties of the form of the vibrational wave
functions leads to a picture of the vibrating diatomic molecule (a picture that is strikingly
different from the classical harmonic oscillator for low-energy vibrations but becomes
more analogous to the classical harmonic oscillator for high-energy vibrations). Let us
now consider how to visualize the features of quantum mechanical features of a vibrating
diatomic molecule, the quantization of the vibrational energy levels, and the wave-like
character of the vibrations. This visualization of χ can be conveniently achieved by
starting with a classical potential energy curve (Fig. 2.3), imposing quantization on the
energy levels (Fig. 2.4), and then describing the appearance of the vibrational wave
functions of the quantized energy levels (Fig. 2.5).
2.18
The Quantized Vibrational Levels of a Harmonic Oscillator
In quantum mechanics, as in classical mechanics, the starting model for a
vibrating molecule is a harmonic oscillator which obeys Hooke's Law (Eq. 2.23).
Solution of the wave equation (Eq. 2.1) for a harmonic oscillator obeying Hooke's law
reveals that the energy levels are quantized and characterized by a quantum number, v
(Fig. 2.4). The potential energy of a each vibrational level, PEv, is given by Eq. 2.27,
where ν is the vibrational quantum number (which can take only integral values, 0, 1, 2,
...), ν is the vibrational frequency of the classical oscillator, and h is Planck's constant.
PEv = hν(v + 1/2)
(2.27)
The important results derived from the quantum mechanical solution for the
harmonic oscillator which contrast strikingly with the results for the classical harmonic
oscillator are:
1.
Only the quantized potential energy values. PEv given by Eq. 2.27 are possible
(stable) for the allowed values of the quantum number v for the harmonically
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vibrating molecule, whereas the values of the potential energy for .a classical
harmonic oscillator are continuous and can take on any value depending on the
energy available the system.
2.
The potential energy, PE0, of the lowest possible vibration is not zero (as it would
be classically for a vibrating spring at rest), but is equal to PE0 = 1/2 hν, the zero
point vibrational energy of a quantum harmonic oscillator. 3.
For a
harmonic oscillator the vibrational energy levels are equally spaced in units of
hν above the ν = 0 level (whose energy is 1/2 hν).
4.
Because of the inherent, irreducible zero point motion of all quantum particles,
the nuclear motion of a vibration cannot be made to cease, i.e., the nuclei never
stops fluctuating about the equilibrium position, so that their average kinetic
energy, KE, can never equal zero. Zero point energy and zero point motion is an
essential feature of every quantum particle, since a quantum particle never is at
rest but will always vibrate (or oscillate) to a certain degree.
Figure 2.4 modifies the picture of the classical harmonic oscillator (Figure 2.3) by
showing the form of the wave functions for each value of v on the corresponding energy
level whose energy is Ev. In Fig. 2.4 the energies corresponding to representative values
of v are indicated by horizontal lines (for v = 0, 1, 2, 3, 4, and 10) for the two situations
depicted in Figure 2.3 for the classical system. The amplitude (extent of the motion of a
representative point that following the vibration from one extreme to the other) of the
corresponding classical vibrational motion is obtained in Figure 2.3 from the intersection
of the potential-energy curve with the corresponding energy level. The classical turning
points of a vibration are the positions defining the maximum potential energy during a
period of vibration. At the turning points, the total energy of the oscillator is potential
energy, because the two masses have stopped vibrating in one direction and are starting
to vibrate back in the other direction. At the kinetic energy is therefore zero at the
classical turning points. Because the representative point is moving slowly near the
turning points, the classical harmonic oscillator tends to spend most of its time at the
turning points. It is important to note that the total vibrational energy, Ev (Eq. 2.27) is
constant during the stretching and compression motions of the nuclei, but the kinetic
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energy, KE, and potential energy, PE, are continually changing so that their sum equals
(Eq. 2.26).
Figure 2.4 Classical potential energy curve with quantized levels superimposed. See
text for discussion.
2.19 The Vibrational Wave Functions for a Harmonic Oscillator. Visualization of
the Wavefunctions for Diatomic Vibrations.
The pictorial forms of the vibrational wave functions for a harmonic oscillator
near the geometry of equilibrium separation are very non-classical and therefore, very
non-intuitive. Thus, inspection of some pictures of vibrational wavefunctions will provide
the student with some “quantum intuition” of how to visualize vibrational wavefunctions
that will be very valuable in the interpretation of both radiative and radiationless
transitions between vibrational levels of states (Chapter 3). The mathematical form of the
vibrational wave functions χv for v = 0, 1, 2, 3, 4 and 10 are plotted qualitatively in
Figure 2.5 (left) on the allowed (quantitized) energy levels, which are indicated by
horizontal lines. Recall that any wave, for example a sine wave, has an amplitude that
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continuous oscillates from (mathematically) positive to (mathematically) negative values.
The (mathematical) value of χv above the horizontal line is arbitrarily considered to be
positive; the (mathematical) value of χv below the horizontal line is negative; and the
(mathematical) value of χv on the line is zero, i.e., the horizontal line is a nodal line for
the wavefunction for all levels above v = 0. The number of times that χv passes through
zero equals v, the vibrational quantum number. It is important to note that the
wavefunction for v = 0 does not possess a node. In this sense the wavefunction is
analogous to the familiar wavefunction for a 1s electronic orbital, which does not possess
a node. The mathematical sign of χv corresponds to the phase of the wave function at
some point in space, r, during a vibration. The sign of χv is crucial when we consider
that χv, a wave function, is not directly related to laboratory observation, but that a
product of two wave functions is related to laboratory observation.
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Figure 2.5. Quantum mechanical description of a vibrating diatomic molecule. See
text for discussion.
Recall (Section 2.4) that the square of a wavefunction represents the probability
of finding particles (electrons, nuclei) in space. Thus, function χν2 (Fig. 2.5, right)
represents the probability of finding the nuclei at a given value of r during vibration in a
given level, i.e., a given value of v. We note that as one goes up the vibrational ladder,
the wavefunction tends to have its highest density near the turning points of the classical
vibration. In this limit the quantum vibration behaves similarly to the classical harmonic
oscillator which spends most of its time at turning points. By examining Figure 2.5
(right) it can be seen that there is a finite, non-negligible probability that the atoms will
vibrate outside of the region defined by the classical potential-energy curve. This
peculiarity is the result of superimposing the wave description of the vibration and the
classical picture. The wave description allows the nuclei to "leak out" of the boundary of
the classical potential energy curve and explore a region of space greater than the
classical description would allow.
Another peculiarity of the quantum mechanical model is apparent through
inspection of the probability of finding the nuclei at a given separation for upper
vibrational levels (Figure 2.5, right). In addition to the broad maxima of the probability
distribution in the vicinity of the classical turning points that is expected from the
classical model, a number of maxima (for ν > 1) exist in between. For example, this
means that for v = 1, there is a low probability of finding the vibrating atoms near re,
whereas for v = 2, there is a relatively high probability of finding the vibrating atoms near
re. Although the behavior of the harmonic oscillator is "peculiar" relative to the classical
model for small values of ν, as we go to higher and higher values of ν, the quantum
mechanical situation approximates the classical situation more closely (Bohr's
correspondence principle), i.e., the atoms tend to spend more time near the turning points
of vibration and very little time in the region about r = re, a situation similar to that
expected for a classical vibration, i.e., A classical representative point has maximum
kinetic energy near re and minimum kinetic energy at turning points.
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The probability distribution curve for the ν = 0 level in Figure 2.5 contrasts most
dramatically with the classical picture. Instead of two turning points (as expected from
the classical model), there exists one broad probability maximum at r = re. Classically,
the vibrational state of lowest energy corresponds to the state of rest (point re in Fig.
2.16). This situation is impossible for the quantum mechanical model, since the position
and velocity of this state would then be exactly defined (a violation of the uncertainty
principle); as a result the vibrating quantum system always possesses a certain amount of
zero point motion. We shall learn in Chapters 4 and 5 that radiative and radiationless
transitions in condensed phases originate from thermally equilibrated vibrational states,
which means that the structure of the ν = 0 level will be of great importance in organic
spectroscopy and photochemistry.
Based on the above descriptions we have built up an important “quantum
intuition” based on the quantization of vibrational energy levels and on visualizing the
wavefunctions of the quantum harmonic oscillator shown in Fig. 2.5. We can now
transfer comfortably and smoothly from the classical description of a "dynamic"
representative point moving on the surface to the quantum description of a "static"
probability of finding the nuclei in a certain region of space with a given motion and
phase. In Chapter 3 we shall employ these ideas to generate a model for an
understanding of the factors determining the probability of transitions between
vibrational levels of different electronic states.
2.20
A First Order Improvement of the Harmonic Oscillator Model. The
Anharmonic Oscillator
The harmonic oscillator is a Zero Order approximation for a vibrating diatomic
molecule X-Y (and for the vibrations of an organic molecule). However, the simple
model fails for nuclear geometries corresponding to severe compressions and especially
for severe elongations of the X-Y bond. In these cases the restoring force does not obey
Eq. 2.23. For example, when a bond has been extended to, say, 2 or 3 times its normal
length (from 1-2 Å to 5-6 Å), the atoms X and Y experience very little "restoring" force,
i.e., the bond is essentially broken and the atoms may fly apart and the restoring force is
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zero for longer separations after the bond has broken. For an ideal harmonic oscillator,
the restoring force increases indefinitely and smoothly with increasing or decreasing
distance from the equilibrium position (see Figs. 2.3, 2.4 and 2.5). For a real molecule,
however, the potential energy will rise more gradually than predicted by PE = 1/2kr2 (Eq.
2.24) because of a weakening of the X-Y bond at large r, as is shown in Figure 2.6 for the
HCl diatomic molecule. The “anharmonic oscillator” takes these factors into account in
the First Order corrections to the Zero Order harmonic oscillator model. For the
anharmonic oscillator as the X-Y bond stretches, the potential energy reaches a limiting
value (the PE of the bond or the bond energy), the restoring force disappears, and the HCl
bond breaks. Thus, the energy of the asymptote in Figure 2.6 corresponds to the
dissociation energy of the molecule. If the energy of the system just corresponds to the
asymptote, the atoms at a great distance from one another will have zero velocity. A
point above the asymptote corresponds to a situation for which the kinetic energy (which
is essentially continuous) of the two atoms is increased. On the other hand, compression
of the nuclei results in a more rapid increase in potential energy than is predicted by the
harmonic oscillator, because of the sudden rise of electrostatic repulsions with decreasing
nuclear separation.
Recall (Eq. 2.26)that the sum total vibrational energy of the system equals the
sum of the kinetic and potential energy. For example, in the ν = 3 vibration level, the
anharmonic oscillator shown in Figure 2.6 possesses 40 kcal/mole-1 at a point A (all
potential energy), while at point B the system possesses about 35 kcal/mole-1 of kinetic
energy and 5 kcal/mole-1 of potential energy (relative to an arbitrary energy of zero for
the lowest point in the curve).
Another important feature of the anharmonic oscillator is that its vibrational
levels, although quantitized, are not equally separated in energy, which contrasts with the
situation for the harmonic oscillator (Fig. 2.4). The energy separations decreases slowly
with increasing ν, as shown in Figure 2.6. For example, for HCl the energy separation
between ν = 0 and ν = 1 is about 12 kcal/mole-1, while the energy separation between ν =
10 and ν = 11 is only about 5 kcal/mole-1.
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Figure 2.6.
2.21
Quantized potential energy curve for HCl.
Summary. Building Quantum Intuition for Vibrational Wavefunctions
Although understanding and use of the paradigm of quantum mechanics requires
a sophisticated mathematical background, the wave functions of electrons, spins and
vibrations may be approximated by pictorial representations that are useful for
understanding the qualitative aspects of organic photochemistry. In this chapter we have
obtained some quantum intuition concerning the behavior of the wavefunctions of
vibrating molecules through quantizing the energy levels of the harmonic oscillator and
visualizing the vibrational wavefunctions. In Chapter 3 we shall use this quantum
intuition to examine how vibrational wavefunctions influence the rates of transitions
between vibrational states of different electronic states.
2.22
Electron Spin. A Vector Model for Visualizing Spin Wavefunctions
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In this Chapter we have seen how the orbital description of electrons allows a
visualization the electronic wave function (ψ) of the location of electrons in space and
that provides quantum intuition for examining electronic issues such as orbital overlap
and features of interactions of electrons that are important in determining the ranking of
electronic energies corresponding to different electronic configuration and estimation of
the electron exchange energy. We have also developed a model of vibrational wave
functions (χ) that allows a visualization of energy levels of vibrating nuclei with the
harmonic oscillator or a Zero Order approximation. This model provides quantum
intuition concerning vibrations of molecules near the zero point (v = 0) level and will be a
powerful means of appreciation of both radiative transitions (Chapter 4) and
radiationless transitions (Chapter 5)between electronic states.
In addition to visualizing the orbital nature of electrons and vibrating nuclei, we
also need to visualize the nature of the electron’s spin wavefunction, S (Eqs. 2.7 and 2.8),
and to develop a working description of the spin structure of molecules that is needed for
an understanding of the mechanisms of transitions between singlet states (Sn) and triplet
states (Tn). Since electron spin is a topic only briefly covered in elementary texts, we
shall develop a relatively detailed model of the electron spin wave functions, S, that is as
pictorial as the orbital model, Ψ, and the harmonic oscillator model for the vibrational
wave function, χ. This model of electron spin, which is based on the properties of
vectors, will provide use with quantum intuition concerning the properties of the spin
vibrational wavefunction, S. It is of interest to note that the language of the vector model
for electron spin is readily transferable for the description of nuclear spins, the key
objects of interest for NMR spectroscopy.
Electron spin is considered in quantum mechanics to be a manifestation of a
fundamental property of electrons (just as the mass and charge of an electron is a
fundamental property), namely the inherent spin angular momentum of an electron. Our
visualization of electron spin will employ the concept of classical angular momentum,
which is the property of a macroscopic object which is in a state of rotation about an axis.
As we have done previously, in order to develop a quantum mechanical picture of
electron spin angular momentum (when the word spin is used we mean spin angular
momentum and its implied magnetic and energetic characteristics to be discussed below),
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we first appeal to a classical model of angular momentum and then modify this model
with the appropriate quantum characteristics.
Two important classes of classical angular momentum involve (1) the circular
“orbital” motion of a particle rotating at a fixed distance about a center and (2) the
circular “spin” motion of a rotating sphere (or a rotating cylinder) about a defined axis
(arbitrarily termed the z-axis). We shall return to case 1 later in this Chapter (when we
deal with electron orbital angular momentum), but we shall first deal with case 2, which
is relevant to electron spin. Taking a concrete physical model of an electron as a sphere
of negative electricity, the spin of an electron is the angular momentum resulting from
its spin motion about an arbitrary axis. Although a classical spinning electron could be
imagined to possess different values of angular momentum depending on the velocity and
direction of its spinning motion, quantum mechanics considers electron spin as an
intrinsic angular momentum which is a fixed and fundamental characteristic of an
electron, similar to its fixed and fundamental charge or mass. Specifically, an electron
possesses a fixed and characteristic amount of spin angular momentum (or spin, for
brevity) of exactly (1/2)h (h, Planck's constant, h, divided by 2π, is the fundamental
quantum mechanical unit of angular momentum) whether it is a "free" electron not
associated with any nucleus, or whether the electron is associated with a nucleus in an
atom, a molecule, an electronically excited state, or a radical. The spin angular
momentum of an electron is always the same, exactly1/2 h, whatever orbital the electron
happens to occupy. Thus, an electron in the n-orbital of a n,π* state has the same spin as
the electron in a π* orbital of a n,π* state. Both have a spin of 1/2 (when dealing with
spin, S, the units of h will be assumed).
When two or more electron spins interact, according to the laws of quantum
mechanics, only certain values of the magnitude of the net spin angular momentum (S)
are allowed. These values are given by S = [S(S + 1)]1/2 (units of angular momentum
understood), where S is the spin quantum number and may be 0, a positive integer or 1/2
a positive integer. In photochemistry the most important values of the spin quantum
number S are 0, 1/2 and 1, corresponding to singlet states, S (0 spin angular momentum),
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doublet states, D (1/2 unit of spin angular momentum) and triplet states, T (one unit of
spin angular momentum).
We have seen (Section 2.7) that electronically excited states, *R, possessing the
same orbital configuration, may exist with different spin configurations of the orbitally
unpaired electrons, i.e., a singlet spin configuration, S1, in which the electron spins are
"antiparallel"(↑↓) for which there is no net spin and a triplet spin configuration, T1, in
which the electron spins are "parallel" (↑↑) and for which there is a net spin of 1 h. We
now show how to visualize the "antiparallel” and "parallel" configurations of two
electron spins by appealing to the classical vectorial properties of an object executing
a spinning motion about an axis, such as a top or a gyroscope. We shall review the
fundamental classical properties of vectors that are important for an understanding of
electron spin and then describe the quantum mechanics of spin in terms of this vector
description.
Since we are interested in the relative energies associated with different spin
configurations and states, we will find it convenient to describe the interactions between
electron spin as those between spin magnetic moments (µ) that we shall directly relate
to the spin angular momentum (S). The idea behind this tactic is that we can build an
intuition for the energy of interactions of two magnets and relate that energy to the
interactions of two spins interacting. It should be born in mind however, that angular
momentum is the fundamental property of electron spin and magnetism is a convenient
framework to discuss the measurable properties of spin qualitatively and quantitatively.
For simplicity, we shall use the same symbol, S, to describe the spin wavefunction and
the quantity, the spin angular momentum.
2.23
A Vector Model of Electron Spin
Some physical quantities are termed scalars if they can be completely described
by a magnitude , i.e., a single number and a unit. Examples of scalar quantities are
energy, mass, volume, time, wavelength, temperature, and length. However, in order to
describe certain physical quantities in addition to a magnitude we must also specify a
directional quality. Quantities which require both a magnitude and a direction for
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complete characterization are termed mathematically as vectors. Examples of vector
quantities are velocity, electric dipoles, angular momentum, magnetic fields, and
magnetic dipoles (also termed magnetic moments). When dealing with spin we shall
represent scalar quantities in italics and vectors in bold face type. For example, spin
angular momentum is a vector quantity and is given the bold face symbol S, whereas the
magnitude of the spin angular momentum is a scalar quantity and is given the italics type
symbol, S.
Both angular momentum and magnetic moments are vectors and may be
completely characterized by a length and a direction. It is natural, therefore, to seek a
vectorial representation of electron spin and to apply vectors to visualize spin angular
momentum S and the magnetic moments µ resulting from spin angular momentum. The
interactions between magnetic vectors will determine the magnetic energies of spin state
possessing two of more interacting electron spins.
When a physical property can be characterized by a vector, the powerful and
simple shorthand rules and algebra of vector mathematics can be employed to describe
and analyze the physical quantities in a completely general manner independent of the
specific quantity the vectors represent. Thus, the same rules of vectors apply to electron
spins and nuclear spins, to angular momentum, to electric dipoles and to magnetic fields.
In contrast to the physical space about the nuclear framework occupied by electrons, it is
useful to think of vectors as existing in a "vectorial space", i.e., independent of the three
dimensional space occupied by the physical object, although it is convenient to place the
vectors in a coordinate system which corresponds to a coordinate system that pertains to
the molecule x, y, z axis. Vector notation will assist in visualizing the wavefunctions for
electron spin and providing quantum intuition for spin in the same way that 3-D
representations of molecular orbitals assist in the visualization of electronic wave
functions.
2.24
Definition and Important Properties of a Vector.
We shall review briefly some properties of vectors that will be use repetively in
description of the properties of electron spin and interacting electron spins. Vectors are
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conveniently represented by an "arrow”. The arrowhead of a vector indicates the sense of
the direction relative to a reference axis (conventionally termed the Cartesian z-axis) and
the length of the arrow represents the magnitude of a physical quantity represented by the
vector.
The orientation of the vector in space is defined in terms of the angle, θ, that the
vector makes with the z-axis. Interacting vectors must all relate to the same axis system.
If the vectors do not interact, each may have its own arbitrary axis system.
2.25
Vector Representation of Electron Spin
Let us consider some important properties of all vectors, with a vector
representation of spin angular momentum S as a concrete exemplar. Figure 2.7
summarizes the well trigonometric relationships of a spin vector, S, that makes an angle θ
with the z-axis. The important vectorial properties are : (1) S possesses a component, SZ
on the z-axis: (2) S possesses a component SX,Y in the x,y plane; and (3) the magnitude
of the component on the z-axis is related to the magnitude of the spin by the
trigonometric Eq. 2. 28, where |SZ| is the magnitude (absolute value) of the vector on the
z-axis and |S| is the magnitude of the spin vector.
cosθ = |SZ|/|S|
(2.28)
The uncertainty principle allows only one component (SZ or SX,Y) of the spin
vector to be specified exactly (conventionally the z-component is the component
selected) the position (termed the azimuth of the vector) of the vector in the xy-plane is
completely undetermined. This is why only the z axis is employed and specified when
discussing the energies of interacting spin states.
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Figure 2.7. Trigonometric relationships between the spin vector, S, and the z axis.
We shall now consider the vector representation of quantized electron spin in
some detail for the two most important cases in photochemistry: a single spin and two
interacting spins which be coupled as parallel (↑↑) or antiparallel spins (↑↓). We shall
the latter description is essentially two dimensional and that for the 3D representation of
interacting spins, these two cases requires a modification of the terms “parallel” and
“antiparallel” when referring to spin configurations of singlets and triplets.
2.26
Vector Model of a Quantized Single Electron Spin. Spin Multiplicities
As mentioned above, the uncertainty principle of quantum mechanics requires
(1) that an exact measurement of the electron spin can be made only of one component
(on the z-axis), leaving the values of the electron spin in the x,y plane completely
unknown and (2) that the values of S and (therefore of SZ) are quantized. The value of
SZ is exactly 1/2 h (values of S will always be in units of h). Since the vector S
representing the spin cannot lie exactly on the z-axis (for then its exact value on all three
axis would be known precisely, in violation of the uncertainly principle), the magnitude
of S must be somewhat greater than 1/2 h and the vector representing the spin must make
a precise angle θ with the z-axis so that the value of SZ becomes exactly 1/2 h on the zaxis. The electron is imagined to be a spinning spherical electrically charged mass that
generates angular momentum as a result of its rotational motion. In Figure 2.8 we show
the angular momentum vector pointing perpendicular to the plane of rotation of a
spinning spherical electron, a feature based on analogy to classical angular momentum of
a spinning sphere.
Recall that quantum mechanics requires that the value of the spin angular
momentum vector be limited to magnitudes determined by the relationship S =
[S(S+1)]1/2, where S is the spin quantum number. The spin quantum number, S, of a
single electron is 1/2 (S is a pure number). Thus, according to spin quantum mechanics,
the actual value of the length S for a single electron is given by (1/2(1/2 + 1)1/2 = (3/4)1/2.
There is only one angle, θ, for which the value of SZ on the z-axis will be exactly 1/2 if
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the length of the vector is (3/4)1/2. From Eq. 2.28, the angle θ that the vector S must make
with the z-axis in order that SZ is exactly 1/2 is about 710 (i.e., from Eq. 2.28, SZcosθ = S,
(1/2) cos71ο = (3/4)1/2). Because of the trigonometric relationships of vectors, the
vectorial representation of spin results in the peculiar "square root" relationship between
the spin and its component on the z-axis. The length of a single electron spin S is a fixed
quantity independent of its orientation, since quantum mechanics requires the vector to
maintain a fixed angle to the z-axis. It should be noted that any particle with spin 1/2
possesses the same value of angular momentum, namely 1/2 hbar. Thus, electrons, 1H
nuclei and 13C nuclei all possess the same value of spin angular momentum, exactly 1/2 h.
Figure 2.8. Vector representation of a spin 1/2 particle (an electron, a proton, a 13C
nucleus). The symbol α refers to the spin wave function of a spin with Ms= +1/2 and
the symbol β refers to the spin wave function of a spin with Ms = -1/2. See text for
discussion.
The number of quantum mechanically allowed orientations of a given state of spin
angular momentum consisting of more than one electron is termed the multiplicity of the
state. The multiplicity of a spin state is computed from the spin angular momentum
quantum number, S according to Eq. 2.26.
Spin Multiplicity = 2S + 1
(2.29)
There are three spin situations of greatest importance to organic photochemistry:
(1) S = 0; (2) S = 1/2; and (3) S = 1. From Eq. 2.29 the multiplicity for S = 0 is 1
(termed a singlet state, S); the multiplicity for S = 1/2 is 2 (termed a doublet state, D);
and for S = 1 the multiplicity is 3 (termed a triplet state, T). We have employed the terms
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singlet and triplet earlier in the text and shall now show their relationship to the number
of possible spin states through the vector model.
It is conventional and convenient to term the spin wave function corresponding to
the quantum number MS = +1/2 is given the symbol α, and the spin wave function
corresponding to MS = -1/2 is conventionally symbolized as β. We shall use the terms α
spin and β spin in reference to these wave functions and consider the vector
representation in visualizing the wave functions.
2.27
Vector Model of Two Coupled Electron Spins. Singlet and Triplet States
We now consider the important interaction or coupling of two single spins, a
situation that is important in describing an electronically excited state *R (or reactive
intermediate I), which possesses two orbitally uncoupled electrons. The spin
configurations of two electrons in singly occupied orbitals are not restricted by the Pauli
Principle, i.e., a singlet or a triplet configuration are both allowed. The quantum
mechanical rule for the possible couplings of two 1/2 spins of the two orbitally uncoupled
electrons is very simple: The final total angular momentum (units of h) of a two spin
system is either 0 (singlet, S) or 1 (triplet, T).
We now shall use the vector model of spin to visualize the singlet and triplet
states that result from the coupling of two electron spins which occupy separate orbitals.
The singlet state results from the coupling of an α spin and a β spin in such a way that the
spin vectors are antiparallel (commonly depicted as ↑↓) and collinear and the angle
between the heads of the vectors is 180o (Figure 2.9a). Such an orientation of vectors
clearly results in exact cancellation of the net spin angular momentum of the spin vectors,
so the net spin vector length is 0. This means that the net spin of the singlet state, as
expected, is exactly equal to 0. Although a specific arbitrary orientation of the two spin
1/2 vectors is shown in Figure 2.9a, there is no preferred orientation at all since all
orientations yield zero spin angular momentum. So, we conclude that the singlet state
has no net spin and no preferred orientation of the component spin vectors, even though it
is composed of two electrons each of which possesses a spin of 1/2. This situation is
analogous to the cancellation of dipole moments in a linear molecule such as O=C=O
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which has two polar bonds, but no net dipole moment. All electrons that are paired in an
orbital are required by the Pauli Principle to be in a singlet state and therefore will have
their spin vectors oriented as shown in Figure 2.9a. As a result, since nearly all organic
molecules possess two electrons in each orbital in their ground states, organic molecules
typically possess singlet ground states.
Figure 2.9. Subtraction and addition of two spin 1/2 particles. Upon subtraction, the
spins exactly cancel (a). Upon addition, the spin vectors add up (b) to a total length of
(2)1/2, but have a projection (right) of 1, 0 and -1 on the z-axis (c).
The triplet state (for which S = 1) also results from the coupling of an α spin and
a β spin, but this time the coupling is such that the spin vectors make an angle between
the vectors which is 71o (Figure 2.9b). In Section 2.27 we shall see that the angle of 71 o
is required by trigonometry so that the two spin 1/2 vectors will couple vectorially to
produce a single spin equal to a vector with a value of S = 1 on the z-axis. According to
quantum mechanics the values of the spin on the z-axis for a S = 1 system can be Sz = +1
h Sz = 0 h or Sz = -1 h . These spin states are assigned the quantum numbers MS = +1,
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MS = 0 and MS = -1 (the symbol M is derived from the magnetic properties of the
magnetic dipole associated with oriented spins), respectively, and are termed the
sublevels of the triplet state. The origin of the term “triplet state” is based on the
quantum mechanical requirement that a S = 1 spin state must possess three energy
sublevels corresponding to the three quantum numbers, MS = +1, MS = 0 and MS = -1.
From trigonometry (Eq. 2.28) the angle θ that the vector must make with the z-axis to
achieve the values of +1 h , 0 h or –1 h are 45o, 90o and 135o, respectively (Figure 2.9c).
Now we can see clearly how the vector model and trigonometry provide a vivid
means of visualization of singlet and triplet spin states based on the rules of quantum
mechanics. For S = 0, the coupled spins exactly cancel and there is no net spin anywhere
in space and only one spin state, a singlet state (quanutum number MS = 0) results. For S
= 1, the coupled spins add to one another and there are three spin states allowed by
quantum mechanics corresponding to the three quantum numbers MS = +1, MS = 0 and
MS = -1.
You will recall that the spin wave function α describes an “up” spin and that the
spin wave function β describes a “down” spin. Notice that both the singlet state and the
triplet component with MS = 0 possess one α spin and one β spin. Thus, a simple two
dimensional up-down "arrow" notation (↑↓) would make the singlet state (MS = 0) and
the MS = 0 level of the triplet appear to be identical in terms of spin characteristics, even
though the singlet has no net spin and the MS = 0 level of the triplet possesses a net spin
of 1h! It is important to realize that this distinction between the MS = 0 for the singlet
and triplet is made only in the three dimensional vector representation which we shall
describe in the next section. This situation of the failure of the two dimensional
representation of spin to distinguish the MS = 0 if the triplet state from the singlet state is
somewhat analogous to the failure of two dimensional representations of molecular
structure to distinguish two enantiomeric centers in molecules.
Now we have to deal with an interesting mathematical modification of our α and
β label of spins for the singlet state and the MS = 0 level of the triplet state that is
imposed by the Pauli Principle and electron exchange (Section 2.9). Because of the
latter, we cannot label the wavefunctions of the two states with one spin up and one spin
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down as either α1β2 or β1α2, as this would imply that we can distinguish electron 1 and
electron 2 as being in a specific and differentiable spin state. This violates the Pauli
Principle which states that electrons are indistinguishable upon exchange of coordinates.
Quantum mechanics allows that an acceptable spin wave function for the singlet state is
(α1β2 − β1α2) where the minus sign signifies that the two spins are exactly 1800 out of
phase (a mathematical “normalization” factor of (1/2)1/2 is of no interest for our
qualitative discussion and is therefore ignored) Αn acceptable spin wave function for the
triplet state is (α1β2 + β1α2), where the plus sign signifies that the two spin are in phase
and couple to generate a net spin of 1. Thus, we will let the singlet state, S, be
represented by the symbols αβ − βα and the triplet state, T, will be represented by the
symbols αβ + βα. Both of these states have the same quantum number MS = 0, but
possess different wavefunctions.
We can think of the minus sign in the singlet function αβ − βα as representing the
"out of phase" character of the two spin vectors causing the spin angular momentum of
the individual spin vectors to exactly cancel (Fig. 2.9a) and the plus sign in the triplet
wave function as representing the "in phase" character causing the individual spin vectors
to add together and reinforce each other (Fig. 2.9 b).
There is no ambiguity for the T+(αα) the triplet with two spins up (αα or MS =
+1) or for T-(ββ) the triplet state with two spins down (MS = -1). For the state with MS =
+1, the spin function αα is acceptable because, since both electrons possess the same
orientation, we need not need to attempt to distinguish them upon exchange. The same
holds for the MS = -1, for which the spin function ββ is acceptable to the Pauli Principle.
2.28
The Uncertainty Principle and Cones of Possible Orientations for Electron Spin
1/2
So far we have discussed the spin vectors in terms of a two dimensional
representation, and now proceed to a more realistic model of the spin vector in 3-D
relative to the z-axis. As we have seen, according to the Uncertainty Principle of
quantum mechanics, if the value of SZ on the z-axis is measured precisely, the 3D
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position of S, i.e., its directional angle of S in the x,y plane (termed the azimuthal angle)
cannot be determined at all. Thus, when SZ is measured there is no information at all that
can be obtained experimentally concerning the azimuthal angle of S. In terms of the
vector model, this means that an up spin, which makes a specific angle of ca 71o with the
z-axis and a value of SZ = 1/2, may project to any azimuthal angle in the x,y plane. Thus,
there is a set of positions in space making an angle of ca 71o with the z-axis, any one of
which could correspond to the actual position of the spin vector. This set of possible
positions constitutes a cone such that whatever the specific position the spin vector
takes on the cone, the angle of the vector with the z- axis and the projection of the
vector on the z-axis are always the same, 710 and 1/2 ; however, the x and y components
or the vector are completely undetermined. Such a cone is termed the cone of possible
orientations of an α spin (MS = +1/2). Figure 2.10 (left top) shows the cone of possible
orientations of an α spin with one possible arbitrary orientation being shown explicitly.
Although we can be sure that an α spin will lie in this cone, we cannot specify where. A
corresponding cone of possible orientations (Figure 2.10, left middle) exists for a β spin
with one possible arbitrary orientation being shown explicitly.
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Figure 2.10 Cones of possible orientation for a spin 1/2 (left) and spin 1 (right) system
of angular momentum. An arbitrary position of the spin vectors is shown for each of
the possible cones. See text for discussion of the insert.
In the absence of an interacting magnetic field, the vector representing the angular
momentum is imagined as being stationary and "resting" somewhere in the cone of
possible orientations. If a coupling magnetic field is applied (e.g., a static applied
laboratory field, or an oscillating applied laboratory field, or fields due to the magnetic
moments generated by the motion of spins in the environment, etc.) the coupling will
cause the spin vector to sweep around the cone, a motion of the spin that we shall refer to
as precession. We shall consider spin precession in Chapter 3 when we consider
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transitions between magnetic states. For now we will only describe the “static” magnetic
states and the energies associated with them.
2.29
Cones of Possible Orientations for Two Coupled 1/2 Spins
In photochemistry *R is generally a singlet state or a triplet state possessing an
electronic configuration that is approximated by two singly occupied orbitals such as a
n,π* or a π,π* configuration. In these cases the two electron 1/2 spins can be viewed as
being coupled to produce a net spin of 1 (triplet) or a net spin of 0 (singlet).
Let us now consider the possible orientations in the case of the triplet of S = 1
(Figure 2.10, right). The situation with respect to a cone of possible orientations of the
spin 1 is analogous to that for the spin = 1/2 case in the cases of MS = +1 (αα) and MS =
-1 (ββ), for which "up" and "down" cones of possible orientations are shown in Fig. 2.10.
In the case of MS = 0 (αβ + βα), the spin vector lies not in a cone, but in a circle of
possible orientations lying in the x,y plane (Figure 2.10, right middle). An interesting
feature of this case is that the projection of the spin on the z-axis is 0 h , even though the
length of S is 1.7 h . It should be noted that the magnitude of the spin vectors for S = 1
are larger (1.7 h) than those for S = 1/2 (0.87 h), although by convention the units are
not usually explicitly shown in the figures. We also note that for the singlet state, strictly
speaking since the value of the spin vector is zero, there is no cone of orientation because
the vector length is zero. However, it is convenient when discussing two weakly coupled
spin 1/2 systems to consider a cone of possible orientations that is determined by the two
1/2 spins (Fig. 2.10)
Finally, we can employ the cone of possible orientations to represent two coupled
1/2 spins to make the point mentioned above concerning the requirement that two 1/2
spins must be precisely in phase in order to produce a spin 1 state. In this case there are
two requirements to make the eventual value of the spin on the z-axis exactly equal to 1:
(1) the angle between the two spin 1/2 vectors must add vectorially to yield a resultant
vector of length equal to 1.7 h and (2) the angle that this resultant vector makes with the
z-axis must be 450, 900 or 1350 in order for the projection of the vector on the z-axis to
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be exactly +1 h, 0 h, or –1 h, respectively. It is important to note that the downward
rotation of the vector from the MS = 1 state by 450 produces the MS = 0 state and a change
of exactly 1 unit of spin angular momentum (ΔS = 1). Similarly, downward rotation of
the MS = 0 state by 45o produces the MS = -1 state and a change of exactly one unit of
angular momentum. Changes in angular momentum of spin states must occur with
conservation of angular momentum and typically occur with a change of one unit of
spin angular momentum (which is exactly compensated by a change in angular
momentum of exactly one unit in the coupled system).
Up to this point in the text we have used the term “parallel” spins (↑↑) to describe
two coupled 1/2 spins in a triplet state. To have the appropriate resultant, two coupled
spin 1/2 systems must make an angle of ca 71 0 with one another (Fig. 2.10) We now see
that the term “parallel” spin actually refers to the condition that the spins make and an
angle of 710 with one another. Importantly, it should be noted that the spins for the αα,
ββ and αβ + βα are all at an angle of 710. Thus, only loosely speaking are the two spins
are parallel for each case.
2.30
Summary
The angular momentum of a rotating or spinning electron or group of coupled
electron spins is conveniently and mathematically represented by a vector S which
according to quantum mechanics must have a length of exactly, [S(S + 1)]1/2 where S is
the electron spin quantum number of the system. Following the rules of quantum
mechanics, the vector representing the angular momentum due to electron spin can only
possess certain quantized values and any particular observable value corresponds to
specific orientations of the spin vector in space. Since only one component of a spin
vector (conventionally the z-axis) can be observed experimentally according to the
uncertainty principle, the azimuth of the vector (its orientation in the xy plane, called its
azimuth) is completely unknown. However, the spin vector must be oriented in one of
the cones of orientation at the specific angles allowed by quantum mechanics.
The vector model for spin angular momentum allows a clear visualization of the
coupling of spins to produce different states of angular momenta. Chemists are
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accustomed to discussing states in terms of their energies and not their angular
momentum. We now need to connect the vector model of angular momentum with
models that allow us to deduce the magnetic energies and how these energies are related
to the orientations of the spin vector in space. Then we shall be in a position not only to
rank the magnetic states in terms of their energies (a goal of this Chapter) but also to
understand the dynamics of transitions between magnetic states, (a goal of Chapter 3).
2.31
The Connection between Angular Momentum and Magnetic Moments. A
Physical Model for an Electron with Angular Momentum.
We shall now develop a physical model for electron spin concentrates on the
magnetic properties and the magnetic energies associated with spin angular momentum.
Electrons and certain nuclei (1H, 13C), possessing both electrostatic charge and spin
angular momentum, also possess magnetic moments (i.e., magnetic dipoles, which are
vectors analogous to electric dipoles) that interact with each other and with applied
laboratory magnetic fields. We will now examine the connection between spin angular
momentum and the magnetic moments associated with spin. We shall consider first apply
the vector model to a familiar physical model of a magnetic moment resulting from
orbital motion of an electron in a Bohr orbit. This concrete physical model allows us to
compute the magnetic moment of the orbiting electron in terms of its orbital angular
momentum. With this connection between orbital angular momentum and magnetic
moments, we shall see how this result can be used to generate a connection between spin
angular momentum and magnetic moments.
We have seen how to qualitatively rank the relative energy of electronic orbitals
and electronic states based on the Aufbau principle and Pauli Principle (Section 2.7). We
also learned how to qualitatively rank the relative energies of vibrational levels based on
their quantum number and number of nodes in their wavefunctions (Section 2.2.17). It
would clearly be useful to develop an model which allows us to rank the relative energy
of the electronic spins and spin states. The importance of magnetic moments is that the
energy of their interactions with a magnetic field can be computed and ranked in terms of
a magnetic energy diagram, analogous to the electronic state energy diagram for
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orbitals (and the vibrational energy diagram for vibrations). Therefore, through the
magnetic moments associated with spin through the application of a magnetic field, we
will be able to construct magnetic energy diagrams ranking the energy of the various spin
orientations corresponding to different states of angular momentum and therefore to
different magnetic moments. An important property of such magnetic energy diagrams is
that, in contrast to the orbital and vibrational energy diagram, the separation between
the energy levels will depend on the strength of the applied magnetic fields which
interact with the electron spin’s magnetic moment.
We have developed quantum intuition concerning the structure and states of
electron spins through the use of a vector model allowing the ready visualization of spin
angular momentum in a three dimensional vectorial "spin space" and showing how the
vectors reside in cones of possible spin orientation. We now need to obtain some physical
intuition concerning the energies of these magnetic states from classical mechanics.
According to classical mechanics, when a magnetic field is present, each orientation of a
given spin state has a different energy. A goal of this Chapter is to provide a means of
qualitatively estimating the energies of states in the presence and absence of magnetic
fields. We will now use the vector model of spin to associate the spin angular momentum
configurations with magnetic moments and magnetic energies in the same way that we
associate electronic configurations with electronic energies. We also need to develop a
model that will allow us to visualize the interactions and couplings of electron spins
leading to transitions between the magnetic energy levels corresponding to magnetic
resonance spectroscopy and intersystem crossing, which will be the topic of Chapter 3.
2.32
The Magnetic Moment of an Electron in a Bohr Orbit
The Bohr model of electrons orbiting about a nucleus provides an exemplar or
concrete physical (electron as a particle moving in a circular orbit) model for showing
how a magnetic moment (µ) is associated with orbital angular momentum (L) and spin
angular momentum (S). Through the model of the Bohr atom, we will connect the
electron’s orbital angular momentum (L) with the magnetic moment (µL) associated
with orbital motion. After developing this model, we shall use it as a basis for deducing
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the relationship connecting quantum mechanical spin angular momentum (S) and the
magnetic moment (µS) associated with spin motion.
To compute the magnetic motion associated with an electron in a Bohr orbit, we
first suppose that an electron is a tiny point negative charge rotating at constant velocity
in a plane about a fixed axis passing through and perpendicular to the positively charged
nucleus (Figure 2.11, left).
The orbital motion of an electron of charge –e generates a
magnetic field (any charge moving in a circle generates a magnetic field). The magnetic
field generated by the orbiting electron may be represented (Fig. 2.11, right) as a
magnetic dipole (a vector quantity analogous to an electric dipole) with a magnitude
proportional to the angular momentum and a direction perpendicular to the plane of the
orbit. Analogous to this classical picture by virtue of its constant circular velocity and
orbital angular momentum, L, an orbiting Bohr electron generates a fixed magnetic
moment, µL, that can be represented by a vector coinciding with the axis of rotation
(Figure 2.11). This behavior is completely analogous to the classical picture of an
electric current flowing in a circular wire, and producing a magnetic moment positioned
at the center of the motion and perpendicular to the plane of the wire. The direction of
the head of the arrow representing the vector follows the “right hand rule” and points
above the plane of rotation for the direction of motion in the orbit shown.
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Figure 2.11. The vector model for the orbital angular momentum and the magnetic
moment due to an circular motion of an electron in a Bohr orbit. The direction of the
magnetic moment vector is opposite that of the direction of the angular momentum
vector for an electron. The units of L are and the units of µL are J-G-1.
From classical physics, the magnetic moment due to orbital motion of the
electron, µL is directly proportional to the magnitude of the orbital angular momentum,
L. This relationship is the important link connecting the value of angular momentum (in
general) to the magnetic moment. From the model of the electron in the Bohr orbit, the
proportionality constant between L and µL can be shown to be -(e/2m), the ratio of the
unit of electric charge to the electron's mass, so that a simple relationship exists between
the fundamental constants of an electron’s charge and mass and the vectors µL and L as
given by Eq. 2.30.
µL = -(e/2m)L
(2.30)
The proportionality constant (e/2m) reflects a fundamental relationship between
the magnetic moment and angular momentum of a Bohr orbit electron and is, therefore, a
fundamental quantity of quantum magnetism. This constant is termed the magnetogyric
ratio of the electron, and is given a special symbol γe. Thus, Eq. 2.30 may be rewritten as
Eq. 2.31 (γe is defined as a positive quantity).
µL = -γe L
(2.31)
If the electron possesses a single unit of angular momentum (h), the magnitude of
its magnetic moment, µL is defined as exactly equal to (e/2m). This quantity of
magnetism is viewed as the fundamental unit of quantum magnetism for the electron and
is given the special name of the Bohr magneton, as well as the special symbol, µe. Its
numerical value is 9.3 x 10-20 JG-1. When we see the symbol µe we can view it as a
magnetic moment generated by an electron possessing an angular momentum of exactly 1
h.
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The following important concepts can be deduced from Eqs. 2.30 and 2.31:
(1)
The vector representing the magnetic moment, µL, and orbital angular
momentum, L, are co-linear (the point in the same direction and are exactly
parallel so that they are equivalent vectors related by a proportionality factor, γe);
(2)
The vector representing the µL is opposite in direction to that of L (the negative
sign in Eq. 2.31 relating vectors means that the vectors possess orientations 180o
apart);
(3)
The proportionality factor γe reveals that the magnitude of the magnetic moment
due to orbital motion is directly proportional to the charge of the electron and
inversely proportional to its mass.
The reason that Eqs. 2.30 and 2.31 are so important is that they allow us to
visualize both the orbital angular momentum, L, which must be strictly conserved in all
magnetic transitions and the magnetic moment, µL, which provides the physical
interactions determining magnetic energies and "triggering" the occurrence of
radiationless and radiative magnetic transitions which will be important for processes
involving the intersystem crossing of singlets and triplet states.. Figure 2.11 presents a
vectorial description of the relationship of µL and L (the Figure is schematic to the extent
that the sizes of the vectors are unitless and not to any particular scale).
Now we need to deduce how a magnetic moment (µS) is associated with electron
spin angular momentum (S). We shall start with the quantum results deduced from how
the orbital angular momentum (L) is associated magnetic moment (µL) and transfer these
ideas to a rotating charged sphere to model electron spin , and then determine what
quantum mechanical modifications of the model are necessary to apply our vector model
to qualitatively visualize spin.
2.33
The Magnetic Moment Associated with Electron Spin
The electron is the lightest quantum particle of chemical interest and, as such, is
and exemplar quantum particle and will therefore possess many properties that are quite
unusual and unpredictable from observations of classical particles. Nonetheless, there is
a clear and appealing visualization that is possible if we start with a model which
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considers the electron as a particle of definite mass and spherical shape that possesses a
unit negative electric charge distributed uniformly over its surface. The properties of
mass and charge are clearly articulated in this simple physical model. In order to
understand the angular momentum of the electron resulting from spin motion, it is
intuitively natural to assume that since the mass of the electron is fixed and since its spin
angular momentum is quantized, and therefore also fixed, the spherical electron must spin
about an axis with a fixed velocity, v (Figure 2.12) in order to obey the fundamental law
of conservation of angular momentum. In other words, because its mass is fixed, and
since it must possess a quantum mechanically fixed unit of angular momentum, the
velocity of the spinning electron is constant for all electrons. It is quite straightforward
to now apply the results from the Bohr atom relating the orbital angular momentum of an
electron to a magnetic moment due to its orbital motion to infer the relationship of the
spin angular momentum of an electron to a magnetic moment due to its spin motion.
Analogous to a charged particle executing orbital motion, an electron executing
spinning motion also generates a magnetic moment or magnetic dipole. Accordingly, we
expect that as a result of its spinning motion, the charged electron will also generate a
magnetic moment, µs, in analogy to the magnetic moment µL generated by an electron in
a circular Bohr orbit. The issue to be addressed is the determination of the relationship
between µs and S.
It is appealing to start with a direct analogy with the relationship of orbital angular
momentum, L, and the associated magnetic moment, µL (Eq. 2.31). An analogy to the
latter equation would suggest that the magnitude of the magnetic moment of the spinning
electron should be equal to γeS, i.e., the magnetic moment due to spin should be directly
proportional to the value of the spin and the proportionality constant should be γe. This
simple analogy is of the correct form, but it turns out that experiment shows that the
expression is not quantitatively correct. Both experimental evidence and a deeper
quantum mechanical theory have shown that although a direct proportionality exists, for
a free electron a "correction factor" ge must be applied to relate µs and S quantitatively
(Eq. 2.32).
µs = -geγeS
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(2.32)
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In Eq. 2.32, ge is a dimensionless constant called the g factor (or g value) of the free
electron and has an experimental value of close to 2. Thus, the simple model which
attempts to transfer the properties of an orbiting electron to a spinning electron is correct
in qualitative form, but is incorrect by a factor of 2. This factor of 2 is fully justified in a
more rigorous theory of the electron which considers relativistic effects, but this rather
subtle issue is not of concern for the qualitative features of spin of interest in this text.
Figure 2.12 uses the vector model to summarize the relationships between the
spin angular momentum and the magnetic moment associated with electron spin. Figure
2.12 should be compared to the analogous form of the orbital angular momentum and its
associated magnetic moment in Figure 2.11.
From Eq. 2.32 and Fig. 2.12, the following important conclusions can be made
concerning the relationship between electron spin and the magnetic moment due to spin:
(1)
Since spin is quantized and the spin vector and the magnetic moment vectors are
directly related: the magnetic moment associated with spin (µS), as the angular
momentum (S) from which it arises, is quantized in magnitude and
orientation;
(2)
Since the energy of a magnetic moment depends on its orientation in a magnetic
field, the energies of various quantized spin states will depend on the
orientation of the spin vector in a magnetic field;
(3)
In analogy to the relationship of the orbital angular momentum and the magnetic
moment derived from orbital motion, the vectors µs and S are antiparallel (Figure
2.12, left);
(4)
The vectors µs and S are both positioned in a cone of orientation which depends
on the value of MS (Figure 2.12, right).
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Figure 2.12. Vector representation of the spin angular momentum, S, and the
magnetic moment associated with spin, µS. The two vectors are collinear, but
antiparallel.
2.34
Classical Magnetic Energy Levels in an Applied Magnetic Field We have now developed a model that allows the association of a specific value of
a magnetic moment with specific value of spin angular momentum. Our next goal is to
develop a model for the magnetic energy levels associated with different values of spin
for electrons in a magnetic field. Let us consider what happens to the magnetic energy
levels when a magnetic field is applied. As has been our protocol, we shall examine the
results for an exemplar classical magnet and then apply these results to determine how
the quantum magnet associated with the electron spin changes its energy when it couples
with any applied magnetic field. The vector model not only provides an effective tool to
deal with the qualitative and quantitative aspects of the magnetic energy levels, but will
also provide us with an excellent tool for visualizing the qualitative and quantitative
aspects of transitions between magnetic energy levels discussed in Chapter 3.
According to classical physics, a magnetic field is characterized by a magnetic
moment, which is a vector quantity and given the symbol H0. When a bar magnet
possessing a magnetic moment µ, is placed in a magnetic field, H0, a torque operates on
the magnetic moment of the bar magnet and twists it with a force that tries to align the
direction of the moment with the direction of the field determined by H0.
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equation describing the energy relationship of the bar magnetic at various orientations in
H0 is given by equation 2.31, where µ and Hz are the magnitudes of the magnetic
moments of the bar magnet and the applied field in the z direction, respectively.
According to classical mechanics, any orientation of the bar magnet in the magnetic field
is possible. Three important limiting orientations, parallel (0o), antiparallel (90o) and
perpendicular (180o), of bar magnetic relative to the magnetic field are shown
schematically in Figure 2.13.
Figure 2.13. Energies of a classical bar magnet in a magnet field, H0. The curved
arrow indicates the force acting to rotate the magnet to align it with the direction of the
field.
E (magnetic) = -µHZcosθ
(2.33)
In Eq. 2.33 the negative sign before the quantity on the right side of the equation
means that the system becomes more stable if the product of mHZcosθ is a positive
quantity. By definition, the magnitudes (lengths) of the vector quantities µ and HZ are
always positive. So, whether the energy of the system is positive (less stable than the
situation in zero field) or negative (more stable than the situation in zero field) depends
on the sign of cosθ (which is positive for θ between 0o and 90o and negative for θ
between 90o and 180o).
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Let us consider the situation when µ and HZ are two parallel vectors (θ = 0), two
perpendicular vectors (θ = 90o or π/2), or two antiparallel vectors (θ = 180o or π). For
these cases, cosθ = 1, 0 and -1, respectively (Figure 2.13). We can make the important
conclusions that any orientation of the bar magnet with θ between 0 and less than 90o is
stabilizing (the E in Eq. 2.31 is negative relative to the zero of magnetic energy) and any
orientation of the bar magnet with θ between 90o and up to 180o is destabilizing (the E
in Eq. 2.31 is positive relative to the zero of magnetic energy). Importantly, at an
orientation of 90o the magnetic interactions between the applied magnetic field and the
bar magnet are zero (cos90o = 0), i.e., the same as they are in the absence of a field!
2.35
Quantum Magnets in the Absence of Coupled Magnetic Fields
We take the classical model for a bar magnetic in a magnetic field and develop a
model for a “quantum magnet” due to electron spin in a magnetic field. We shall
concentrate on the three most important spin situations that are encountered in molecular
organic photochemistry: a single electron spin (which is found for free organic radicals)
and two coupled spins (which is found for singlet and triplet states).
We establish a convention that will be a convenient form to discuss the important
spin situations of a single spin and two coupled spins. Table 3 summarizes the notations
and vector representations for a single spin and two coupled spins and will be used to
describe spin systems at both zero and high applied magnetic fields. We use the symbol
D (doublet) to describe the state of a single spin. For the case of MS = +1/2 (α or up
spin), we label the state D+; and for the case of MS = 1/2 (β or down spin), we label the
state D-. In the case of two coupled spins we use the symbol S to denote the singlet state
(one α spin and one β spin, wavefunction = αβ + βα). This state has the quantum number
MS = 0. For the triplet state we use the symbols T+ (two α spins), T0 (one α spin and one
β spin, wavefunction = αβ + βα) and T- (two β spins)to label the states with quantum
numbers of MS = +1, 0 -1, respectively. We have noted in Section 2.26 that both the S
and T0 states possess one α spin and one β spin and a quantum number of MS = 0 and
appear to be equivalent. We shall address and remedy this situation below.
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In the absence of magnetic interactions operating on the magnetic moments
associated with electron spins, all of the five states (D, S, T-, T0, T+) have exactly the
same energy, i.e., the magnetic energy levels of all of these states are all degenerate. In
the absence of any magnetic interactions, the vector model of a single electron spin views
the vector representing the angular momentum and magnetic moment as stationary and
possessing random orientations in space. There is no pertinent MS quantum number in
for zero field, because there is no preferred axis of orientation. However, even at zero
field the total spin angular momentum (S = 0, 1/2, 1) is well defined, so we say that the
quantum number for total spin is still a good quantum number. In other words, a D or T
state is magnetic and a S state in non-magnetic whether or not an applied field is present.
2.36
Constructing a Magnetic Energy Diagram. Quantum Magnetic Energy Levels
in a Strong Applied Field. Zeeman Magnetic Energies
The magnetic energy of interaction of a magnetic moment and an applied field
(Figure 2.13) is known as the Zeeman energy. Eq. 2.32 relates the magnetic moment of
the quantum magnet due to electron spin with the value of the g factor, the strength of the
magnetic field and the value of the spin angular momentum. Eq. 2.34 relates the energy
of the quantum magnet to the strength of the applied field, H0, the magnetic quantum
number for a given spin orientation, MS, the g-value of the electron and the universal
constant µe, the magnetic moment of a free electron.
E = MsgµeHZ
(2.34)
Thus, the ranking of magnetic energies (relative to those in zero field) of the
singlet, doublet and triplet states are readily computed and are shown in Table 3.
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State
State
MS
Symbol
Magnetic
Spin
Vector
Energy
Function
Representation
Doublet
D+
+1/2
+(1/2)gµeHZ
α
z axis
Doublet
D-
-1/2
-(1/2)gµeHZ
β
z axis
Singlet
S
0
0
αβ - βα
z axis
Triplet
T+
+1
+(1)gµeHZ
αβ + βα
z axis
Triplet
T0
0
0
αβ + βα
z axis
Triplet
T-
-1
-(1)gµeHZ
αβ + βα
z axis
Table 3. Conventional representations of singlet, doublet and triplet states in an
applied magnetic field HZ. A mathematical normalizing factor is not shown for the
spin function.
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2.37
Magnetic Energy Diagram for A Single Electron Spin and Two Coupled
Electron Spins
Figure 2.14 displays the magnetic energy level diagram at zero magnetic field and
in the presence of a magnetic field for the two fundamental cases of : (1) a single
electron spin, a doublet or D state and (2) two correlated electron spins, which may be
either a triplet, T, or singlet, S state. In zero field (ignoring the electron exchange
interaction, J, and only considering the magnetic interactions) all of the magnetic energy
levels are degenerate, because there is no preferred orientation of the angular momentum
and therefore no preferred orientation of the magnetic moment due to spin. Although
these states are all of the same energy, they are shown with levels slightly separated so
that the number of degenerate states is clear. Remember that these states have identical
energies in the absence of magnetic interactions.
The zero field situation is a point of calibration of magnetic coupling energy in
devising a magnetic energy diagram. The concept is the same as using the energy of a
non-bonding p orbital as a zero of energy and then to consider bonding orbitals as lower
in energy than a p orbital, and anti-bonding orbitals as higher in energy than a p orbital.
The magnitude of the exchange interaction (Section 2.7) is typically much larger than
magnetic interactions. However, the exchange interaction is a Coulombic (electrostatic)
interaction and is not magnetic. This allows us to consider the magnetic interactions first
and independently, as shown in Figure 2.14, and to later "turn on" the electrostatic
interactions after considering the magnetic interactions.
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Figure 2.14. Magnetic energy diagram for a single electron spin and two correlated
electron spins.
2.38
Magnetic Energy Diagrams Including the Electron Exchange Interaction
As mentioned above, the exchange interaction, J, between two electrons results in
a quantum mechanical electrostatic (non-magnetic) splitting of the energy of the singlet
state from the triplet states (Section 2.7). As a result, the energy diagrams shown in
Figure 2.14 must be modified in the presence of electron exchange. First we need to
comment on a significant difference between the situation for singlet-triplet splittings for
molecules (*R and R) and for reactive intermediates (I). Consider a radical pair for
which a single bond has been stretched and broken as an example of I. Fig. 2.6 is an
exemplar potential energy curve for such a situation. Near the minimum of the curve the
electron exchange is strong in both R and *R because the molecular orbitals of X and Y
are close together (rxy ca 0.7 Å) and strongly overlapping. When the bond is stretch to ca
3 Å the bond is essentially broken. At this point overlap between the orbitals of X and Y
is small. Since the value of the overlap integral, J, falls off rapidly as a function of
distance, there is relatively little electron exchange in the region of separation near and
beyond values of separation of ca 3Å and larger. In the case discussed in Section 2.7, the
singlet and triplet states being compared were S1 and T1, both derived from a HO-LU
orbital configuration corresponding to *R. Even in the case of *R = n,π* the value of the
exchange integral is of the order of several kcal/mol or greater.
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Let us use the term T and S for the triplet state and the singlet state of I. For a
radical pair the value of J can be of the order of 10-3 kcal/mol. Such weak exchange
interactions have two important consequences: (1) the magnitude of the exchange forces
between electrons become of the same order of magnitude as the magnetic forces
between electrons for I; (2) T no longer is lower in energy than the S for I. The reason
for (1) is straightforward and has to do with the fall off of J and the magnetic forces with
separation of X and Y. The reason for (2) is a bit more subtle, but is a also
straightforward if we consider that for large separations (infinity) of X and Y which
correspond to J = 0, the singlet and triplet have the same energy! This is the same
situation that would exist if the electrons in molecules did not correlate with one another.
So as X and Y move closer together, what happens to the relative energies of the two
states? Clearly the S0 state is always lower than the T1 state, so it would not be surprising
if for I the S state was lower in energy than the T state. The reason that the S state drops
in energy is that the two electrons with paired spins correlate with S0 which does not
possess an antibonding electron. T on the other hand must correlate with T1, which does
possess an antibonding electron. When comparing * R(S1) with * R(T1) both states
possess an antibonding electron. In summary, the value of J for radical pairs will be
small and the S state of a radical pair will generally be lower in energy that of the triplet
state, T. In these cases the electron exchange effectively stabilizes the system by
contributing to bonding in addition to repulsions when the two electrons are together in
the singlet state. When the orbitals which contain the two electrons of interest are
involved in orbitals for which bonding is substantial, J becomes a negative energy terms
and makes the electron spin pairing situation (singlet) more stable than the spin unpairing
situation (triplet).
There, three limiting situations for which the combination of exchange and
magnetic field effects are important in photochemical systems.
For case I one J = 0 and H0 = 0 (Figure 2.15, left). In Case I although S and T are
distinct states, they are degenerate in energy. Importantly, the three sublevels of T (T+, T0
and T-1) are degenerate (same energy) and can be viewed as rapidly interconverting
along the molecular axis as the molecule tumbles and rotates in solution. This situation
may be considered in analogy to three rapidly equilibrating conformers of identical
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energy and structure. We shall see that Case I is typical of situations for which there is
no magnetic field present and the two spins are separated by large distances (>3-5 Å) in
space and as is the case for solvent-separated, spin-correlated geminate pairs and long
biradicals.
Case II is one for which H0 = 0, but for which there is a finite value, but small
value of J, comparable to the value of the magnetic interactions between the electron
spins. Case II is typical of situations for which the two spins are close enough together so
that their orbitals overlap slightly in space as is the case for spin correlated radical pairs
and small biradicals. Case III is one for which a high field, H0, is present and for which
the values of J are 0 or are comparable to the Zeeman splitting (i.e., J = 0 or J ~ gµeH).
In this special case, the T- and S state may have comparable energies. This situation is
important as we shall see in Chapter 3, because closeness in energy sets up favorable
conditions for transitions between states. In other words, when T- and S are very close in
energy, the conditions are favorable for T- → S and S → T- intersystem crossings in I.
Figure 2.15 Effect of electron exchange on the magnetic energy levels of a triplet
and singlet state. See text for discussion
2.39
Interactions between Two Magnetic Dipoles. Orientational and Distance
Dependence of the Energy of Magnetic Interactions
A magnetic moment is a magnetic dipole, i.e., the magnetic moment gives rise to
a magnetic field in its vicinity Fig. 2.16. We shall investigate the mathematical form of
the two interactions in order to obtain some intuition concerning the magnitude of the
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interactions as a function of spin structure and orientation of spins in space. Fig. 2.16
shows one of the line of force which emanate from a magnetic dipole (the same picture,
importantly, holds for electric dipoles). The lines of force can be traced as leaving and
then reentering the vicinity of the dipole. The “direction” of the magnetic force varies as
a function of the space about the dipole. For example, in regions 1 and 3 the lines of
force are in the same direction as that of the dipole moment. However, in region 2 the
lines of force are in an opposite direction to that of the dipole moment. Clearly the
degree of interaction between two dipoles will depend on their relative orientation to one
another.
Figure 2.16. (a) A dipole (magnetic or electric); (b) one line of force the force field in
the vicinity of a dipole; (c) a dipole-dipole interaction with parallel dipoles at an angle
of 90 o to one another; (d) a dipole-dipole interaction with parallel dipoles at an angle
of 0 o to one another.
Insight to the nature of the magnetic dipole-dipole interaction is available from
consideration of the mathematical formulation of the interaction and its interpretation in
terms of the vector model. The beauty of the formulation is that its representation
provides an identical basis to consider all forms of dipole-dipole interactions. These may
be due to electric dipoles interacting (two electric dipoles, an electric dipole and a nuclear
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dipole or two nuclear dipoles) or to magnetic dipoles interacting (two electron spins, an
electron spin and a nuclear spin, two nuclear spins, a spin and a magnetic field, a spin and
an orbital magnetic dipole, etc.).
Classically, the dipole-dipole interaction energy depends on the relative
orientation of the magnetic moments (consider two bar magnets). To obtain some
concrete insight to the dipolar interaction consider the case for which the two magnetic
dipoles, µ1 and µ2, are held parallel to one another (this is the case for two interacting
magnetic dipoles in a strong magnetic field (Fig. 2.18). We can obtain an intuitive
feeling for the nature of the dipole-dipole interaction by considering the terms of the
dipole-dipole interaction in Eq. 2.35. In general, the strength of the interaction is
proportional to several factors: (1) the magnitudes of the individual interacting dipoles;
(2) the distance separating the interacting dipoles; (3) the angles of orientation of the
dipoles relative to one another; and (4) the "spectral overlap" of resonances that satisfy
the conservation of angular momentum and energy. In fact, Eq. 2.33 strictly speaking
refers to the interaction of two "point" dipoles (if r, the separation between the dipoles is
large relative to the dipole length, the dipole may be considered a "point" dipole).
Dipole-dipole interaction a [(µ1µ2)/r3](3cos2θ - 1)(overlap integral)
(2.35)
For dipole-dipole interactions in solution, the rate of processes involving dipoledipole interactions is typically proportional to the square of the strength of the dipoledipole interaction. Thus, the field strength falls off as 1/r3, but the rate of a process
driven by dipolar interactions falls off as 1/r6.
The term involving the 3cos2θ- 1 (Fig. 2.17) is particularly important because of
two of its features: (1) for a fixed value of r and interacting magnetic moments, this term
causes the interaction energy to be highly dependent on the angle θ that the vector r
makes with the z axis and (2) the value of this term averages to zero if all angles are
represented, i.e., the average value of cos2θ over all space is 1/3. A plot of y = 3cos2θ- 1
is shown at the bottom of Figure 2.17. It is seen that the values of y are symmetrical
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about a = 90 o. It is interesting to note that for values of a = 54o and 144o, the value of y
= 0, i.e., for these particular angles, the dipolar interaction disappears even when the
Figure 2.17. Dipole-dipole interactions of parallel magnetic moments. Top: vector
representation of dipoles interacting at a fixed separation, r, and various orientations
relative to a z axis. Bottom: plot of the value of 3cos2θ - 1 as a function of q.
spins are close in space! This is the familiar "magic angle" employed to spin samples in
the magnetic field of an NMR spectrometer for removing chemical shifts due to dipolar
interactions. It is important to note that certain values of y are positive (magnetic energy
raising) and certain values of y are negative (magnetic energy decreasing).
2.39
Summary
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We have developed working paradigms for describing the structure and energetics
of electron spin employing a vector model of spin and relating it to magnetic moments.
In Chapter 3 we shall consider the dynamics of spin that are critical in processes which
interconvert spin states, in particular the intersystem crossing between spin states
involved in singlet triplet interconversions and triplet singlet interconversions.
Although understanding and use of the paradigm of quantum mechanics requires
a sophisticated mathematical background, in this chapter we have attempted to describe
and picture the wave functions of electrons, spins and vibrations may be approximated by
pictorial representations that are useful for understanding the qualitative aspects of
organic photochemistry. In this chapter we have considered the visualization of the
electronic, spin and vibrational structure of the starting points for photochemical
reactions, R*. In Chapter 3 we will consider the visualization of the photophysical and
photochemical transitions of R*.
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